Algebra and Trigonometry Enhanced with Graphing Utilities (7th Edition)

Chapter 6

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; Put  in the equation  and solve for  to find ... more

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Yes, the expression is polynomial. The degree of ... more

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down; 4. ; The graph of touches the x-axis at . ... more

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Local maxima is and local minima is . ; Plot the ... more

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The real solutions of the equation  are the values... more

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The point on the graph of  is . The x-intercept is... more

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continuous; smooth. ; The graph of every ... more

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touches ; If the multiplicity of a real zero  of a... more

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; The graph of function always contains the ... more

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is a factor of the polynomial;is a real zero/root ... more

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Turning points. ; The point at which a function ... more

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; The graph of is a U-shaped curve as shown ... more

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; As  becomes large,  the function  approximates ... more

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The notation  means that, as  approaches to large ... more

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The degree is the highest power in the polynomial ... more

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The number of turning points can not be negative... more

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is a polynomial function of degree 3. The standard... more

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is a polynomial function of degree 4. The standard... more

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The function  is a polynomial function of degree, ... more

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The function  is a polynomial function of degree 1... more

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The function  is not a polynomial function due to ... more

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The function  is a polynomial function of degree, ... more

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The function  is not a polynomial function, ... more

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The function  is not a polynomial function, ... more

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The function  is a polynomial function of degree, ... more

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The function  is not a polynomial function, ... more

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The function  is a polynomial function of degree, ... more

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The function  is a polynomial function of degree, ... more

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Use the graph of .  ; Shift the graph of to the ... more

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Use the graph of .  ; Shift the graph of to right... more

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Use the graph of . ; Shift the graph of down by 3... more

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Use the graph of .  ; Shift the graph of up by 2 ... more

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Use the graph of .  ; Compress the graph of in ... more

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Use the graph of . ; Stretch the graph of ... more

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Use the graph of . ; Reflect the graph of about ... more

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Start with the graph of . Plot the three key ... more

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Start with the graph of . Plot the three key ... more

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Start with the graph of . Plot the three key ... more

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Start with the graph of . Plot the three key ... more

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Start with the graph of . Plot the three key ... more

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Plot the three key points, , , and . The graph ... more

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Start with the graph of . Plot the three key ... more

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; If is a zero with multiplicity of a ... more

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; If is a zero with multiplicity of a ... more

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; If is a zero with multiplicity of a ... more

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; If is a zero with multiplicity of a ... more

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; If is a zero with multiplicity of a ... more

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; If is a zero with multiplicity of a ... more

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; If is a zero with multiplicity of a ... more

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; If is a zero with multiplicity of a ... more

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; Since the zeros are , the factors are , , and... more

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; Since the zeros are , the factors are , , and... more

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; Since  the zeros are , the factors are , , and... more

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; Since the zeros are , the factors are , , and... more

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; Since the zeros are and 1, the factors are ... more

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; The zeros of are (multiplicity 2), 0, and 3 (... more

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Could be a polynomial.Real roots: , and Least ... more

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Could be a polynomial.Real roots: and Least ... more

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The graph can not be the graph of a polynomial ... more

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The graph can not be the graph of a polynomial ... more

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; The graph has -intercepts, which represent real... more

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; The graph has -intercepts, which represent real... more

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; The graph has -intercepts, which represent real... more

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; The graph has -intercepts, which represent real... more

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; The graph has -intercepts, which represent real... more

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; The graph has -intercepts, which represent real... more

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; The graph has -intercepts, which represent real... more

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; The graph has -intercepts, which represent real... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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; The graph of the polynomial function looks ... more

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; Since the leading term of the polynomial ... more

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; Since the leading term of the polynomial ... more

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; Since, the leading term of the polynomial ... more

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; Since, the leading term of the polynomial ... more

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; Since, the leading term of the polynomial ... more

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; Since, the leading term of the polynomial ... more

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; Since, the leading term of the polynomial ... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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The end behavior of the function is .The x-... more

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 or  ; If is a zero of multiplicity of a ... more

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; If is a zero of multiplicity of a polynomial ... more

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; If is a zero of multiplicity of a polynomial ... more

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; If is a zero of multiplicity of a polynomial ... more

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Start by graphing the given function. As ... more

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The graphing of a polynomial function requires to ... more

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; The graph of a polynomial crosses the x-axis ... more

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,  . ; The graph of a polynomial crosses the x-... more

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Smooth but not continuous: ,Continuous but not ... more

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True. Every polynomial intersects the y-axis only ... more

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.The polynomial is not unique. ; Choose four ... more

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noyes, which yields the -intercept of a function... more

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; Rewrite the given equation in slope-intercept ... more

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; The function has denominator 0 at . ; Since, ... more

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; For the function , the roots are given by the ... more

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; The equation is equivalent to the two ... more

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; Substitute  for  in the equation for . ; ... more

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; Find . is the coefficient of and is the ... more

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Quotient:  Remainder:   ; Use synthetic division ... more

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; Start by setting to 0. ; Since the trinomial ... more

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In , is the remainder. ; In , is the dividend... more

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; The Remainder Theorem states that if is ... more

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In , the sign changes from to . Then, there is ... more

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Since the degree of the polynomial is , the ... more

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; is of the form with . Since is a factor of , ... more

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The dividend is is a polynomial of degree . The ... more

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The remainder is .  is a factor of . ; Determine ... more

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The remainder is . is not a factor of .  ; Write... more

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The remainder is . is a factor of . ; Determine ... more

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The remainder is . is a factor of . ; Determine ... more

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The remainder is . is a factor of . ; Write in ... more

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The remainder is . is a factor of .  ; Write in ... more

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The remainder is . is not a factor of . ; Write ... more

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The remainder is . is a factor of . ; Write in ... more

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The remainder is . is a factor of . ; Determine ... more

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The remainder is . is not a factor of . ; Write ... more

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The maximum number of real zeros is . There are ... more

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The maximum number of real zeros is . There is ... more

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The maximum number of real zeros is . There are ... more

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The maximum number of real zeros is . There is ... more

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The maximum number of real zeros is . There are ... more

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The maximum number of real zeros is . There are ... more

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The maximum number of real zeros is . There is ... more

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The maximum number of real zeros is . There are no... more

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The maximum number of real zeros is . There are... more

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The maximum number of real zeros is . There is ... more

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The maximum number of real zeros is . There are no... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; List all the integers . These are the factors ... more

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; Determine the maximum number of zeros by ... more

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LB: ; UB: 1 ; List the potential rational zeros of... more

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LB: ; UB: 1 ; List the potential rational zeros of... more

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LB: ; UB: 11 ; List the potential rational zeros ... more

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LB: ; UB: 3 ; List the potential rational zeros of... more

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LB: ; UB: 3 ; List the potential rational zeros of... more

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LB: ; UB: 3 ; List the potential rational zeros of... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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; Using the Descartes' Rule of Signs, identify ... more

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The signs of and are opposite. Then, there is at... more

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The signs of and are opposite. Then, there is at... more

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The signs of and are opposite. Then, there is at... more

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The signs of and are opposite. Then, there is at... more

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The signs of and are opposite. Then, there is at... more

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The signs of and are opposite. Then, there is at... more

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Find the – intercept by evaluating when . The... more

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Find the – intercept by evaluating when . The... more

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Find the – intercept by evaluating when . The... more

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Find the – intercept by evaluating when . The... more

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Find the - intercept by evaluating when . The... more

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Find the – intercept by evaluating when . The... more

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Step 1: Step 2:Step 3:Step 4: ; Step 5:Step 6: ; ... more

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Step 1: Step 2:Step 3:Step 4: ; Step 5:Step 6: ; ... more

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; Evaluate when .  ; If is a factor of , then... more

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; Evaluate when .  ; f is a factor of , then . ... more

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The remainder is . ; Use the Remainder Theorem. ... more

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The remainder is . ; Use the Remainder Theorem. ... more

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Since, then is a factor of . , then is a factor ... more

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Since , then is a factor of . ; Let. Check if is... more

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The sum of the remaining solutions is . ; Divide ... more

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The sum of the remaining solutions is . ; Divide ... more

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The length of the edge of a cube is. ; Let be the ... more

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The length of the edge of a cube can be or ; Let ... more

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All the potential zeros are integers. If is in the... more

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is a factor of the constant term and is a factor ... more

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The zero of correct to three decimal places is. ; ... more

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The graph is increasing on the open intervals and... more

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Sum: Product: ; Write the sum of the two complex ... more

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, ; This function does not factor, so use the ... more

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one ; This is the corollary that follows the ... more

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; By the conjugate pairs theorem, if  is a zero, ... more

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The theorem that follows the Fundamental Theorem ... more

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By the Conjugate Pairs Theorem, if  is a zero, ... more

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; By the conjugate pairs theorem, if  is a ... more

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; By the Conjugate Pairs Theorem, if  is a ... more

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, ; By the Conjugate Pairs Theorem, if  is a ... more

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; By the Conjugate Pairs Theorem, if  is a ... more

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, ; By the Conjugate Pairs Theorem, if  is a ... more

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; By the Conjugate Pairs Theorem, if  is a ... more

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; By the Conjugate Pairs Theorem, if  is a ... more

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, ; By the Conjugate Pairs Theorem, if  is a ... more

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, ; By the Conjugate Pairs Theorem, if  is a ... more

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, , ; By the Conjugate Pairs Theorem, if  is a ... more

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; According to the Conjugate Pairs Theorem, if  ... more

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; According to the Conjugate Pairs Theorem, if ... more

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; If is a zero of , then is also a zero of . ... more

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; If is a zero of , then is also a zero of . ... more

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; If is a zero of , then is also a zero of . ... more

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; If is a root of , then is also a root of . ... more

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, 4  ; If is a zero of , then is also a zero of... more

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,  ; If is a root of , then is also a root of . ... more

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,,  ; If is a root of , then is also a root of... more

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,,  ; If is a root of , then is also a root of... more

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,,  ; If is a root of , then is also a root of... more

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; If is a root of , then is also a root of . ... more

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, , ,  ; If is a root of , then is also a root ... more

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, -3, 4,  ; If is a root of , then is also a ... more

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Zeros: Factored form of : ; Factor  using the ... more

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Zeros: , , , Factored form of :  ; Factor using ... more

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Zeros: , , 2Factored form of : ; List the ... more

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Zeros: -5, , Factored form of :  ; List the ... more

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Zeros: , , , Factored form of :  ; Note that this ... more

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Zeros:, , , Factored form of :  ; Note that this ... more

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Zeros: Factored form of :  ; List the potential ... more

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Zeros: Factored form of :  ; List the potential ... more

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Zeros: Factored form of :  ; List the potential ... more

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Zeros: Factored form of :  ; List the potential ... more

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130 ; Since , plug this into and solve for .  ; ... more

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; Since is a zero of , is also a zero of . ... more

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If is a zero, then is a zero. If is a zero, ... more

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If  is a zero, then  must also be a zero. But  is ... more

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Since already has 3 given zeros, there is only ... more

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Since already has 3 given zeros, there is only one... more

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The given theorem only applies to polynomials with... more

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Plot each point on the coordinate plane. ; more

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-22 ; Square both sides of the equation. ; Solve... more

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Area = square feetCircumference = square feet ; ... more

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A rational function is a function of the form  ... more

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Quotient:  ; Remainder: ; Divide  by . ; Determine... more

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Write the equation in TI-83 calculator for ... more

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Graph the function  using a transformation. The ... more

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The domain of the every rational function is the ... more

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Horizontal asymptote ; It is a horizontal line ... more

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Vertical asymptote ; It is vertical lines which ... more

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Proper ; The result is true for all rational ... more

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If the end behavior of the graph occurs as  or as... more

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If the end behavior of the graph occurs when  is ... more

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; If the degree of the numerator  is less than ... more

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If the degree of the numerator  is equal to the ... more

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If the degree of the numerator  is less than the ... more

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If, as  approaches some number , the values , then... more

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; Set the denominator equal to zero, and solve ... more

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; Set the denominator equal to zero and then ... more

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; Set the denominator equal to zero and then ... more

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; Set the denominator equal to zero and then ... more

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; Set the denominator equal to zero and then ... more

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; Set the denominator equal to zero and then ... more

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; Set the denominator equal to zero and then ... more

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. ; Set the denominator equal to zero and then ... more

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. ; Set the denominator equal to zero and then ... more

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. ; Set the denominator equal to zero and then ... more

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; Set the denominator equal to zero and then ... more

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. ; Set the denominator equal to zero and then ... more

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The rational function has a horizontal asymptote... more

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Vertical asymptote: .Horizontal asymptote: .... more

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Vertical asymptote: .Horizontal asymptote: No ... more

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Vertical asymptote: and .Horizontal asymptote: No... more

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Vertical asymptote: and .Horizontal asymptote: .... more

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Vertical asymptote: .Horizontal asymptote: .... more

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Vertical asymptote: .Horizontal asymptote:  .... more

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Vertical asymptote: No vertical asymptote.... more

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Vertical asymptote: .Horizontal asymptote: No ... more

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Vertical asymptote: .Horizontal asymptote: No ... more

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If is the real zero of the denominator of the ... more

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If the degree of the denominator is equal to the ... more

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A horizontal asymptote will occur only when the ... more

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.  ; The oblique asymptote is .Let us assume that... more

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; Determine the equation of a vertical line ... more

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.  ; Determine the value of x from the given ... more

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Symmetric with respect to -axis. ; Determine ... more

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; Determine the point of intersection of the ... more

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The intercepts of the graph are the points... more

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If the degree of the numerator is 2 more than the ... more

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A horizontal asymptote may intersect the graph of ... more

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Set and solve for y to find the -intercept.    ; ... more

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These are the zeros of the denominator which makes... more

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 2. R is in lowest terms 3. -intercept:  NONE, -... more

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 2. R is in lowest terms 3. -intercept: , -... more

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 2. R is in lowest terms 3. -intercept:  , -... more

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 2. Q is in lowest terms 3. -intercept:  , -... more

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 2. R is in lowest terms  4. R is in lowest terms... more

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 2. R is in lowest terms 3. -intercept: -1; -... more

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 2. P is in lowest terms 3. -intercept : -1, -... more

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 2. Q is in lowest terms 3. -intercept:  , -... more

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 2. R is in lowest terms 3. -intercept: ; -... more

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          2. R is in lowest terms 3. -intercept... more

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 2. G is in lowest terms 3. -intercept: ; -... more

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 2. G is in lowest terms 3. -intercept:  , -... more

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 2. R is in lowest terms 3. -intercept: ; -... more

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 2. R is in lowest terms 3. -intercept:  , -... more

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 2. H is in lowest terms 3. -intercept: ; -... more

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 2. H is in lowest terms 3. -intercept:  , -... more

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 2. F is in lowest terms 3. -intercept: ; -... more

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 2. F is in lowest terms 3. -intercept:  , -... more

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                                                  ... more

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 2. R is in lowest terms 3. -intercept:  , -... more

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 2. F is in lowest terms 3. -intercept: ; -... more

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 2. G is in lowest terms 3. -intercept:  , -... more

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 2. R is in lowest terms 3. -intercept: ; -... more

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 2. R is in lowest terms 3. -intercept:  NONE, -... more

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                     2. In lowest terms,  3. -... more

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                  2. In lowest terms,  3. -... more

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                                                  ... more

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                                                  ... more

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                             2. In lowest terms... more

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 2. In lowest terms,  3. -intercept: ; -intercept... more

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 2. In lowest terms,  3. -intercept:  , -intercept... more

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 2. In lowest terms,  3. -intercept:  , -intercept... more

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 2. In lowest terms,  3. -intercept:  , -intercept... more

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 2. G is in lowest terms 3. -intercept:  , -... more

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 2. G is in lowest terms 3. -intercept:  , -... more

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 2. f is in lowest terms 3. -intercept:  NONE, -... more

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 2. f is in lowest terms 3. -intercept:  NONE, -... more

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 2. f is in lowest terms 3. -intercept:  NONE, -... more

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 2. f is in lowest terms 3. -intercept:  NONE, -... more

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 2. f is in lowest terms 3. -intercept:  NONE, -... more

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 2. f is in lowest terms 3. -intercept:  NONE, -... more

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The possible equation is    .        ; A rational ... more

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; Let be a rational function. The -intercepts of... more

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The possible equation of the graph is ; The -... more

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As the number of increases, increases. Thus, a ... more

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is not a valid test point. A test point has to be... more

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,  ; Graph the function, ; The graph has ... more

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, ; Graph the function, . ; The graph has ... more

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,  ; Graph the function, . ; The graph has an ... more

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, ; Graph the function, . ; The graph has an ... more

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; Subtract from both sides of the inequality and... more

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; Subtract from both sides of the inequality and... more

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; Subtract from both sides of the inequality and... more

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; Subtract from both sides of the inequality to ... more

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; Subtract from both sides of the inequality to ... more

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; Add to both sides of the inequality and ... more

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; Subtract from both sides of the inequality to ... more

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; Add to both sides of the inequality to get .... more

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; Subtract from both sides of the inequality and... more

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; Subtract from both sides of the inequality and... more

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c ; Determine the graph’s end behavior by ... more

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47,105 ; Evaluate . Substitute 4 for in .Perform ... more

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The upper bound is not found. The lower bound is... more

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and none ; is already in lowest terms, no need to ... more

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none ; Simplify to lowest terms.Factor the ... more

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is in lowest terms.The -intercept is .The -... more

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1.  2.  3. The -intercept is .The -intercept is .... more

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in lowest terms and the domain are given here... more

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