Algebra and Trigonometry: Real Mathematics, Real People (7th Edition)

Chapter 4

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The graph of the function is a parabola that ... more

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The function is a parabola that opens in the ... more

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; The vertex of the function is given as . ... more

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; Compare the equation of the path of the ... more

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The value of  changes the direction in which the ... more

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Non Negative Integer ; Consider the following ... more

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Quadratic Function ; A polynomial function ,  with... more

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Yes ; Simplify the function.The highest degree of ... more

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Yes ; Simplify the function.The highest degree of ... more

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The graph of parabola opens downward and it has ... more

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The graph of parabola opens downward and it has ... more

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The graph of parabola opens downward and it has ... more

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The graph of parabola opens downward and it has ... more

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Sketch the graph of the equation   with vertex ... more

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 There is a vertical shift downwards by one unit ... more

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 The obtained graph in red colour is the ... more

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The graph is The function is a reflection of the ... more

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Horizontal shift of 1 unit in the left direction.... more

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The graph is 2 units upward of and is a ... more

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The graph of the given equation is a horizontal ... more

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The graph has a reflection along -axis, a ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Graph of the function is a parabola that opens in ... more

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Parabola, which opens in the upward direction.... more

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Parabola, which opens in the upward direction.... more

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Parabola, which opens in the downward direction.... more

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Parabola, which opens in the downward direction.... more

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Parabola, which opens in the downward direction.... more

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Parabola, which opens in the downward direction.... more

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; The graph has its vertex at . Substitute this ... more

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; The graph has its vertex at . Substitute this ... more

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; Substitute the value of the vertex for  in the... more

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; Substitute the value of the vertex for (h,k) ... more

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; Substitute the value of the vertex for (h,k) ... more

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; Vertex of the function is .Point lying on the ... more

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; Vertex of the function is .Point lying on the ... more

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; Vertex of the function is .Point lying on the ... more

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; Use the method of completing the square to find... more

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; Use the method of completing the square to find... more

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; Use the method of completing the square to find... more

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; Use the method of completing the square to find... more

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; Use the method of completing the square to find... more

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; ; Enter the equation into the graphing utility.... more

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; ; Enter the equation into the graphing utility.... more

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; Use CAS to draw graph of the quadratic function... more

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; Use CAS to draw the function  and represent ... more

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; Draw the graph of the quadratic function and ... more

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  ; Write the equation of the quadratic function ... more

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  ; Write the equation of the quadratic function ... more

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   ; Write the equation of the quadratic function ... more

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; Write the equation of the quadratic function ... more

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55; 55 ; Let the two numbers be and .Find the ... more

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3333 ; Consider two positive real numbers x and... more

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1272 ; Consider two positive real numbers and , ... more

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  ; Consider two positive real numbers and , and ... more

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feet ; Compare the given equation with the general... more

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TrueThe value of can not be negative. So the ... more

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Trueis the only x -intercept point of this ... more

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FalseThe vertex points of both given equations are... more

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True, both equations have same axis of symmetry... more

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The standard form of the parabola is   which open ... more

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The standard equation of a parabola with its ... more

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Graph of the given function is a horizontal shift ... more

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In relation to the graph , the family of parabolas... more

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The family of parabolas represented by can be ... more

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Parabolas represented by is obtained by a ... more

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; Compare the given equation with the standard ... more

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; Compare the given equation with the standard ... more

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; Compare the given equation with the standard ... more

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; Compare the given equation with the standard ... more

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Since is the sum of and , write in terms of .  ... more

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Let  and be the two real roots. Add both the ... more

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  Solving by hand or using technology, result the ... more

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For the function , when is positive, is minimum ... more

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; Isolate for the equation . ; Substitute for ... more

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; Substitute for into the equation . ; Solve ... more

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; ; Substitute for  into the equation . ; ... more

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; Equate the two given expressions for to solve ... more

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As the time increases, the height of softball ... more

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Function is at the bottom inside the interval .... more

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; In example 2, the degree of the function is, ... more

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The curve falls on the left side and rises on the ... more

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  ; To find the zeros, substitute in the equation... more

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Real zeros are Extremes are ; To find the zeros, ... more

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Real zeros are   ; To find the zeros, substitute ... more

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; The root will correspond to . ;   ; The root ... more

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According to the leading coefficient test, the ... more

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; Calculate the value of the function for various... more

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Continuous ; The graph of a polynomial function is... more

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; The polynomial function of degree n in one ... more

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Touches ; Consider a polynomial function with ... more

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No ; With reference to the given graph, the ... more

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No ; The leading coefficient of function cannot ... more

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The sign of the function changes from negative, at... more

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Repeated zero of in interval has even ... more

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There are three zeros of the graph, while the ... more

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There are three zeros of the graph, while the ... more

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There are three zeros of the graph, while the ... more

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(a) is correct as the coordinates of the graph ... more

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It is a cubic equation graph and the function ... more

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It is a cubic equation graph but not the same. It ... more

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It is a cubic equation graph but not the same. It ... more

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It is a cubic equation graph but not the same. It ... more

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The transformation from to is represented by the... more

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The transformation from to is represented by the... more

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The transformation from to is represented by the... more

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The transformation from to is represented by the... more

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The transformation from to is represented by the... more

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The transformation from to is represented by the... more

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Yes ; Enter both equations, and , into the CAS to... more

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Yes ; Enter both equations, and , into the CAS to... more

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Yes ; Enter both equations, and , into the CAS to... more

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Yes ; Enter both equations, and , into the CAS to... more

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No, as the leading coefficients of both the ... more

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No, as the leading coefficients of and   have ... more

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The graph rises on the left and also to the right... more

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The graph rises on the left and also to the right... more

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The graph rises on the left and also to the right... more

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The graph falls on the left and rises on the right... more

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The graph falls on the left and rises on the right... more

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The graph rises to the right and falls to the left... more

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The graph falls to the left and right. ; The ... more

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The graph falls to the left and right both. ; The ... more

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and ; Substitute zero for in the given function... more

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and ; Substitute zero for in the given function... more

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Zeros: 3; Multiplicity: 2 ; Substitute 0 for in ... more

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Zeros: ; Multiplicity: 2 ; Substitute 0 for in ... more

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Zeros: , 1; Multiplicity: 1  ; Substitute 0 for ... more

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Zeros: 3,4; Multiplicity: 1 ; Substitute 0 for in... more

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Zeros: 0, 2;Multiplicity: 1, 2 ; Substitute 0 for... more

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Zeros: 0, , 5;Multiplicity: 2, 1, 1 ; Substitute 0... more

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Zeros: , ;Multiplicity: 1 ; Substitute 0 for in ... more

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Zeros: , ;Multiplicity: 1 ; Substitute 0 for in ... more

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Zeros: , , , ;Maxima: ;Minima: ,   ; Enter the ... more

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Zero: , Relative maxima: , Relative minima: ; ... more

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Zero: Relative maxima: Relative minima: ; Enter ... more

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Graph the equation using a graphing utility and ... more

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Zero: , , , Relative maxima: , .Relative minima... more

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Zero: , , Relative maxima: Relative minima: ; ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Let be the function of the given zeros. Write ... more

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; Write the factor form of given zero, with ... more

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; Write the factor form of given zero, 3 with ... more

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; Write the factor form of given zero, with ... more

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; Write the factor form of given zero, 5 with ... more

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; Write the factor form of given zero, with ... more

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; Write the factor form of given zero, 1 with ... more

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Let and be the two real zeros of the polynomial... more

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Let , , and be the zeros of the polynomial with ... more

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Zero: , , Relative maxima: Relative minima: ; ... more

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Let us take a polynomial function and check ... more

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No symmetryTwo -intercepts ; Enter the equation ... more

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No symmetryTwo -intercepts ; Enter the equation ... more

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Symmetry with respect to -axisTwo -intercepts ; ... more

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No symmetryTwo -intercepts ; Enter the equation ... more

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Symmetry: originThree-intercepts ; Enter the ... more

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Symmetry with respect to -axisThree -intercepts ; ... more

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No symmetryThree -intercepts ; Enter the equation... more

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No symmetryTwo -intercepts ; Enter the equation ... more

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; Plot the approximate point on the graph of the ... more

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years ; Enter the expression into the graphing ... more

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True, the root can be of a multiplicity of 6. ; A ... more

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False, the graph of a fifth-degree polynomial ... more

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False, the graph of functions with even degree can... more

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True, the degree of the given function is odd and ... more

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False; the graph of the function touches the ... more

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False; the graph crosses the -axis at and . ; ... more

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False; the graph rises to both left and right. ; ... more

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The graph falls to the left and rises to the right... more

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69 ; Substitute the value of and in . ; ... more

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33 ; Subtract the two functions and simplify .  ; ... more

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; Multiply the two functions, with function . ; ... more

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; Divide the function by to find .  ; ... more

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109 ; Use the definition of the composition of ... more

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72 ; Use the definition of the composition of ... more

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; Remove the parentheses. Subtract from both ... more

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or . ; Subtract 1 from both sides of the ... more

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; Subtract by 4 in each part of the inequality . ... more

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  or ; Add 1 in each part of the inequality . ; To... more

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; Multiply by to obtain . Subtract from . ; ... more

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; Multiply by to obtain . Subtract from... more

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; Write the given expression in the descending ... more

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; The given polynomial is divided by to ... more

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; Use synthetic division to divide the given ... more

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; 2 ; 4 ; The leading coefficient of the ... more

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; The leading coefficient is two and the constant... more

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; Find out the variation of sign in the given ... more

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; The possible real zeros can be found by ... more

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; Factor out the common term . ; The possible ... more

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is not the root of the polynomial. ; Let the given... more

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The point on the -axis of the graphs. It means ... more

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Enter the equation into the graphing utility.... more

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is the dividend is the divisor is the quotient is ... more

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Improper  ; Any rational expression of the form  ... more

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Constant termLeading coefficient ; For any ... more

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Descartes’s rule Signs ; Consider to be a ... more

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Upper Lower ; Let be a polynomial with real ... more

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; According to Descartes's rule of sign, if the ... more

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7 ; Let be a polynomial that gives the quotient ... more

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2 ; From the synthetic division method, the terms ... more

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; Multiply by , and then subtract the result ... more

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; Multiply by , and then subtract the result ... more

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; Multiply by , and then subtract the result ... more

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; Multiply by , and then subtract the result ... more

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; Multiply by and subtract the result from . ... more

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; Multiply by and subtract the result from . ... more

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; Multiply by , and then subtract the result ... more

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; Multiply by , and then subtract the result ... more

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; Write the given dividend in descending powers ... more

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; Write the given dividend in descending powers ... more

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; Multiply by , and then subtract the result ... more

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; Multiply by , and then subtract the result ... more

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; Multiply by to get and subtract it from . ; ... more

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; Expand the term . ; Multiply by , and then ... more

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; Set an array of constant terms and write the ... more

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; Set an array of constant terms and write the ... more

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; Set an array of constant terms and write the ... more

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; Set an array of constant terms and write the ... more

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; Set an array of constant terms  and write the ... more

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; Set an array of constant terms and write the ... more

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; Set an array of constant terms and write the ... more

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; Set an array of constant terms and write the ... more

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; Set an array of constant terms and write the ... more

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; Use the synthetic division method to divide by... more

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Graphically, the expressions and are equivalent... more

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Enter the equations into the graphing utility to ... more

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Enter the equations into the graphing utility to ... more

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Enter the equations  into the graphing utility to ... more

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; Use synthetic division to divide the equation ... more

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;   ; Use the synthetic division to obtain the ... more

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; Use synthetic division to divide by ; Write ... more

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; Use synthetic division to divide by .  ; Write... more

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; Use synthetic division to divide by .  ; Write... more

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; Use synthetic division to divide by . ; Write ... more

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; ; 4  ; Divide the equation, by using synthetic... more

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1; 5; ; Divide the equation, by using synthetic ... more

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If the remainder is 0 after applying synthetic ... more

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If the remainder is 0 after applying synthetic ... more

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If the remainder is 0 after applying synthetic ... more

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If the remainder is 0 after applying synthetic ... more

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Possible rational zeros: Rational zeros: ; As the ... more

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Possible rational zeros: Rational zeros: , ; As ... more

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Possible rational zeros: Rational zeros: , , , ; ... more

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Possible rational zeros: Rational zeros: ; As the ... more

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4, 2 or 0 positive real zeros0 negative real zero... more

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3 or 1 positive real zeros1 negative real zero ; ... more

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2 or 0 positive real zeros1 negative real zero ; ... more

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1 positive real zeros No negative real zero ; For ... more

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Yes, 4 is the upper bound and is the lower bound ... more

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Yes, 4 is the upper bound and is the lower bound ... more

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Yes, 5 is the upper bound and is the lower bound ... more

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Yes, 3 is the upper bound and is the lower bound ... more

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; ; The leading term is 4 and the constant term is... more

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;;4 ; The leading term is 2 and the constant term ... more

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; ; The leading term is 4 and the constant term is... more

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;; ; The constant term is and the leading term is... more

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The leading coefficient of is 1 and the constant ... more

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The leading coefficient of is 1 and constant term... more

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Factor out from .Use the identity to factor .... more

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Factor out from .  Rewrite the function as a ... more

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;1; ; Graph of the function intersects the -axis ... more

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; Consider the function given in the question and... more

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; Consider the  function given in the question ... more

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; Consider the  function given in the question ... more

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; Enter the equation into CAS to obtain the ... more

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; Enter the equation into CAS to obtain the ... more

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; Enter the equation into CAS to obtain the ... more

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; Enter the equation into CAS to obtain the ... more

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; Enter the equation into CAS to obtain the ... more

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; Enter the equation into CAS to obtain the ... more

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4 ; The leading coefficient and constant term of ... more

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1 ; The leading coefficient and constant term of ... more

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0; ; ; Take out the common monomial factor . ; One... more

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0; ; 2; ; ; Take out the common monomial factor... more

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1; 2; ; ; ; Write the possible rational zeros of ... more

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1; ; ; ; ; Write the possible rational zeros of ... more

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False, because the zeros of is not it is . ; ... more

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If  is a zeros of the polynomial, that means, the... more

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; From the given graph, it is observed that the ... more

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; From the given graph, it is observed that the ... more

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; From the given information, it can be concluded... more

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; If is a factor of the function, it implies ... more

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; Write the quadratic equation and add 17 on ... more

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; Write the equation and add 1 to both sides. ; ... more

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  ; Compare the equation with the standard ... more

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; Compare the equation with the standard ... more

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; Factorise the expression. ; The expression has ... more

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; Factorise the expression. ; The expression has ... more

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Zeros: 4, , , ; Plot the graph of .The graph ... more

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; Form a function that has roots and . ; The ... more

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; Form a function that has root and . ; The ... more

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; and are the two roots of as complex roots ... more

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Fundamental theoremalgebra ; The fundamental ... more

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Irreduciblereal  ; When a quadratic factor is ... more

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; A polynomial function, , of degree can be ... more

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; As is a root of a fourth degree polynomial, ... more

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Since the degree of is four, the number of zeros ... more

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Since the degree of is five, the number of zeros ... more

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Since the degree of is three, the number of zeros... more

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Since the degree of is one, the number of zeros ... more

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; Factorize the polynomial completely. ; The ... more

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; Factorize the polynomial completely. ; The ... more

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; Factorize the polynomial completely. ; The ... more

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; Factorize the polynomial completely. ; The ... more

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One real root and one x-intercept ; is a cubic ... more

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Three real roots are the three x-intercept , , and... more

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No  real root and no x-intercept. ; is a fourth ... more

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Two real roots are the two x-intercept at and... more

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; Substitute and the resulting zeros of the ... more

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; Substitute and the resulting zeros of the ... more

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; Substitute and the resulting zeros of the ... more

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; Substitute and the resulting zeros of the ... more

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; Substitute and the resulting zeros of the ... more

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; Substitute and the resulting zeros of the ... more

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; Substitute and the resulting zeros of the ... more

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; Substitute and the resulting zeros of the ... more

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; Substitute ,the zeros of the function equals to... more

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; Substitute ,the zeros of the function equals to... more

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; Substitute , the zeros of the function equals ... more

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; Substitute ,the zeros of the function equals to... more

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; appears to be a root of the . ; Use synthetic ... more

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; appears to be a root of the . ; Use the ... more

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; appears to be a root of the . ; Use the ... more

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Zeros: ; The possible rational zeros are and . ... more

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Zeros: ; The possible rational zeros are , , , and... more

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Zeros: ; The possible rational zeros are , , , and... more

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Zeros: ; The possible rational zeros are , , , , ... more

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Zeros: ; The possible rational zeros are , , and... more

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  ; Write the given zeros of the polynomial. ; Now... more

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; Write the given zeros of the polynomial. ; Now... more

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; Write the given zeros of the polynomial. ; ... more

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; Write the given zeros of the polynomial. ; ... more

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; Write the given zeros of the polynomial. ; ... more

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; Write the given zeros of the polynomial. ; ... more

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; Since is a given zero of the given function... more

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

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

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

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

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, , and 3. ; The complex zeros always consist of ... more

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Due to the imaginary value of , football would not... more

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No, there is no price that would yield a profit of... more

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False; a three-degree polynomial always has at ... more

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False; imaginary roots of a polynomial always ... more

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Remainder Theorem:This method is used when the ... more

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The given function is a parabola that opens upward... more

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The given function is a parabola that opens ... more

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The given function is a parabola that opens upward... more

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The given function is a parabola that opens upward... more

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Domain: All real numbers except 1. ; The ... more

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 Horizontal asymptote: ;Vertical asymptotes are ; ... more

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Vertical asymptote: Horizontal asymptote: Hole: ; ... more

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The denominator is zero when , so the domain of ... more

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Function (a) is not continuous.Function (b) is ... more

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The vertical asymptote is . ; Make a table for the... more

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is not a horizontal asymptote. ; Enter the ... more

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Rational function ; , where and are polynomials ... more

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Vertical asymptote ; If as from the left (or ... more

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Vertical asymptote ; By solving , find the ... more

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No  ; In , the degree of the numerator is equal ... more

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By referring to the graph given in the question, ... more

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By referring to the graph given in the question, ... more

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By referring to the graph given in the question, ... more

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By referring to the graph given in the question, ... more

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By referring to the graph given in the question, ... more

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By referring to the graph given in the question, ... more

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Vertical asymptote : Horizontal asymptote : ; To ... more

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