Algebra and Trigonometry (1st Edition)

Chapter 5

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; Vertex is shifted above by and shifted left by... more

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; The general form of the quadratic equation is .... more

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Domain is all real numbers;Range is ; For a ... more

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No intercept;  intercept ; To calculate the ... more

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It is easy to know the coordinates of the vertex ... more

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Building bridges ; The basic example of real value... more

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If , the quadratic function takes the form of a ... more

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Equation of second degree. ; Equation of the ... more

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Factoring and completing the squares method. ; In ... more

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;Vertex ; The given equation is in the general ... more

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; Vertex ; The given equation is in the general ... more

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;Vertex . ; The given equation is in the general ... more

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; Vertex . ; The given equation is the general ... more

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; Vertex ; The given equation is in the general ... more

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; Vertex . ; The given equation is in the general... more

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, Vertex ; The given equation is in the general ... more

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, Vertex  ; The given equation is in the general ... more

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Exists a minimum value.Minimum value:.Axis of ... more

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Minimum value.Minimum value: .Axis of symmetry... more

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Maximum value.Maximum value: 7Axis of symmetry: ; ... more

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Minimum value ; Comparing the equation with the ... more

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Maximum value ; Comparing the equation with the ... more

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Minimum value ; Comparing the equation with the ... more

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Maximum value6 ; Comparing the equation with the ... more

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Domain is Range is ; The domain of the function ... more

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Domain is Range is ; The domain of the function ... more

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Domain is .Range is . ; Comparing the equation ... more

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Domain is .Range is . ; Comparing the equation ... more

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Domain is .Range is  . ; Comparing the equation ... more

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; To find the vertex form of the quadratic ... more

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; To find the vertex form of the quadratic ... more

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; To find the vertex form of the quadratic ... more

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; To find the vertex form of the quadratic ... more

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; To find the vertex form of the quadratic ... more

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; To find the vertex form of the quadratic ... more

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; To find the vertex form of the quadratic ... more

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; To find the vertex form of the quadratic ... more

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The standard equation of a quadratic equation is... more

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The vertex of the graph is located at .The ... more

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• The vertex of the graph is located at .• The ... more

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The vertex of the graph is located at .The ... more

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The vertex of the graph is located at .The given ... more

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The vertex of the graph is located at .The ... more

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; From the curve it is clear that the coordinate ... more

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; From the graph, the coordinates of vertex is . ... more

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; From the graph, it is clear that the ... more

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; From the graph, it is clear that the ... more

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; From the graph, it is clear that the vertex of ... more

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; From the graph it is clear that the vertex is... more

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; Let the quadratic equation be . From the table... more

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; Let the quadratic function be . From the given ... more

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; Let the equation be . From the table, it is ... more

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; Let's assume the equation be . From the table, ... more

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; Let's assume the quadratic equation be . From ... more

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For the same value of , the value of the function ... more

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By adding the constant value, the vertex of the ... more

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When the constant numbers are added with , the ... more

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; Let Graph the equation using a graphing utility... more

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; Let Graph the equation using a graphing utility... more

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; Since the graph of the function is a parabola ... more

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; Since the graph of the function is a parabola ... more

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; The domain of the parabola is Since the graph ... more

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; The domain of the parabola is . Since the graph... more

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; The equation of the quadratic function with ... more

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; The equation of the quadratic function with ... more

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; The equation of the quadratic function with ... more

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; The equation of the quadratic function with ... more

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; The equation of the quadratic function with ... more

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; The equation of the quadratic function with ... more

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; Suppose the length of the rectangle is and the... more

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; Since the rectangle splits into 2 pens of the ... more

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; Since the rectangle splits into 3 pens of the ... more

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33;9 ; Let be two numbers, such that their sum is... more

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6 and -6 ; -36 ; Suppose that are two number such... more

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1800 units ; The revenue is modeled as a parabola ... more

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; The height of the rocket is given by the ... more

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; The height traveled by the ball in air is given... more

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; Let be the average attendance and be the ... more

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Approximately 41. ; Let be the number of trees ... more

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The function can be simplified as Since the ... more

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The function values decrease indefinitely as the ... more

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6, , -1 ; The greatest power in in the functional... more

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as It has the shape of a polynomial with an even ... more

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3  as ,   as ; The general form of a polynomial is... more

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intercept: . intercepts:,  , and  . ; The ... more

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; The polynomial function can be rewritten in ... more

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The degree of the polynomial is odd and at least 3... more

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intercepts at , , and , intercept at , the ... more

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The degree is the largest power of the variable, ... more

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The expression should be expanded first. ; The ... more

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as and as . This implies that for values ... more

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Any  polynomial function of degree can  have ... more

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The polynomial has an even degree and a negative ... more

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The functions and can both be represented by the... more

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. ; The functional expression is , which can be ... more

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. ; The greatest power of the variable in the ... more

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. ; The greatest power of the variable in the ... more

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Degree: 4; Leading coefficient: -2 ; Use the ... more

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Degree: 4; Leading coefficient: 4 ; Use the ... more

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 As ; As ; The degree of the  polynomial is 4 and ... more

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As ;As ; The degree of the function, is 3 and the ... more

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As ;As ; The degree of the polynomial is 4, which... more

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As ;As ; The degree of the polynomial is 9, which... more

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As ;As ; The degree of the polynomial is 4, ... more

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As ; As ; The degree of the polynomial is 2, ... more

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As ;As ; Simplify the polynomial by using the ... more

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As ;As . ; Simplify the polynomial by using the ... more

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intercept: ; intercept: ; Substitute 0 for in the ... more

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intercept: ;intercept: ; Substitute 0 for in the ... more

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intercept:;intercept: ; Substitute 0 for in the ... more

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intercept: ;intercept: ; Substitute 0 for in the ... more

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intercept: ;intercept: ; Substitute 0 for in the ... more

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intercept: ;intercept: ; Substitute 0 for in the ... more

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3 ; Since it is observed that the end behavior of ... more

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2 ; Since it is observed that the end behavior of ... more

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5 ; It is observed that the graph has 5 -... more

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1 ; Since it is observed that the end behavior of ... more

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3 ; It is observed from the graph that it has one... more

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4 ; It is observed from the graph that it has two... more

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5 ; It is observed from the graph that it has one... more

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2 ; It is observed from the graph that it has zero... more

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Yes; one turning point; least possible degree is 2... more

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Yes; zero turning point; the least possible degree... more

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Yes; one turning points; least possible degree is ... more

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Yes; zero turning points; the least possible ... more

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Yes; zero turning points; the least possible ... more

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No ; It is observed from the graph that it has a ... more

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Yes; zero turning points; the least possible ... more

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As , 10-1000100-1,000,000-101000-1001,000,000 ; ... more

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As , .109,50010099,950,000-109,500-10099,950,000... more

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 As and as 10810010098,010,000-1012100-100102,010... more

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As and as 10-504100-941,094-101,716-1001,061,106... more

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As and as 100100900,000,000-10-20,000-100-1,100,... more

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-intercepts are and ; -intercept is ; as and as... more

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-intercepts are and ; -intercept is ; as and as... more

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-intercepts are , and ; -intercept is ; as and as... more

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-intercepts are , and ; -intercept is ; as and as... more

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Draw the graph of the function using the graph ... more

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Draw the graph of the function using the graph ... more

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Draw the graph of the function using the graph ... more

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Draw the graph of the function using the graph ... more

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Draw the graph of the function using the graphing... more

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Draw the graph of the function using the graphing... more

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; Since the degree of the polynomial is 2 and the... more

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; Since the degree of the polynomial is 2 and the... more

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; ; Since the degree of the polynomial is 3 and ... more

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; Since the degree of the polynomial is 3 and the... more

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; The degree of the polynomial is 4. Also, since ... more

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; Since the radius is increasing at the rate of... more

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. ; Since the edge is increasing at the rate of , ... more

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. ; Since the length is increased by , the new ... more

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. ; The dimensions of the cardboard are .Since the... more

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. ; Consider the width of the rectangle to be ; so... more

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intercept :; intercept : ; To find the intercept... more

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The first zero is at and the multiplicity is 1.... more

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Find different values by substituting some ... more

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Substitute in the function and simplify to find ... more

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; It is observed that there are three -intercepts... more

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The minimum value of is 0 and the maximum value ... more

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The zeros of a polynomial can be found by ... more

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The graph of this function will cross the -axis at... more

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The intermediate value theorem, which states that ... more

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A factored form of a polynomial helps to find the ... more

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If the graph of a polynomial  touches the axis ... more

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-intercepts: , and . ; To find the -intercepts of ... more

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The -intercepts are and . ; To find the -... more

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The -intercepts are , and . ; To find the ... more

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The intercepts are and . ; To find the ... more

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The intercepts are and . ; To find the ... more

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The intercepts: , and . ; To find the intercept ... more

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The -intercepts are and . ; To find the -... more

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The -intercepts are , and . ; To find the -... more

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The -intercepts are , and . ; To find the -... more

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The -intercepts are , and . ; To find the -... more

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The intercepts are , and . ; To find the ... more

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; To find the -intercepts of the polynomial ... more

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The -intercepts are and . ; To find the -... more

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  ; To find the intercept:Equate the function to... more

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, , and   ; To find the intercept by equating the... more

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, , and ; To find the intercept, equate the ... more

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, , and ; To find the intercept by equating the ... more

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and ; To find the intercept by equating the ... more

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The intermediate value theorem states that there ... more

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The intermediate value theorem states that there ... more

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The intermediate value theorem states that there ... more

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Polynomial function changes its sign it is confirm... more

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The intermediate value theorem states that there ... more

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The intermediate value theorem states that there ... more

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Zeroes: , 3Multiplicity of : 3Multiplicity of 3: 2... more

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Zeroes: 0, , 4Multiplicity of 0: 2Multiplicity of... more

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Zeroes: , 0, 1Multiplicity of 0: 3Multiplicity of... more

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Zeroes: -2, 0Multiplicity of -2: 2Multiplicity of ... more

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Zeroes: , Multiplicity of : 3Multiplicity of : 2... more

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Zeroes: , 5.Multiplicity of : 5.Multiplicity of 5... more

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Zeroes: 0, , Multiplicity of 0: 1.Multiplicity of... more

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Zeroes: 0, , 2Multiplicity of 0: 4Multiplicity of... more

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Zeroes: 0, Multiplicity of 0: 2Multiplicity of : 2... more

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Multiplicity of is 3.Multiplicity of is 2. ;  ... more

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Multiplicity of is 5.Multiplicity of is 2. ; The... more

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Multiplicity of is 6.Multiplicity of is 2. ; The... more

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x-intercepts: with multiplicity 2, with ... more

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x-intercepts: y-intercept: End-behaviour: , ; The ... more

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x-intercepts: having multiplicity 2, having ... more

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x-intercepts: having multiplicity 3, having ... more

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x-intercepts:   each having multiplicity 1y-... more

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-intercepts: ,, each with multiplicity 1-intercept... more

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; The intercepts are at . At , the nature of the... more

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; The intercepts are at . Therefore there will ... more

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;  The values of intercepts are . Therefore the ... more

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; The values of the intercepts are . Therefore ... more

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; The values of intercepts are , therefore the ... more

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with multiplicity 1  ; From the graph, the values... more

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 , Multiplicity is 1. , Multiplicity is 2. ; From ... more

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Multiplicity is 2. ; -axis is equal to zero for... more

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, Multiplicity  1., Multiplicity 3. ; are the ... more

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; Since the degree of the function is 3, there ... more

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; Since the degree of the function is 3, ... more

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; Root of multiplicity 2 at means the polynomial... more

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; Polynomial has the root of multiplicity 2, ... more

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; Double zero implies a zero of multiplicity 2 ... more

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; Polynomial has zeros at and polynomial has ... more

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; The polynomial has roots and has degree equal ... more

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; Roots have multiplicity 2, hence polynomial ... more

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; Root has multiplicity equal to 2, hence ... more

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; is a zero of multiplicity equal to 2 as it is ... more

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; A local minima or maxima is a smallest or ... more

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Maxima: Maxima: ; Open any graphing calculator and... more

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; Use the graphing utility to plot the graph of ... more

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; Use the graphing utility to plot the graph of ... more

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; Use the graphing utility to plot the graph of ... more

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; From the graph the function has three zeros, ... more

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; From the graph polynomial has two zeroes. One ... more

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; From the graph, the function has two zeroes, ... more

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; Length is equal 10 units, width equal to 8 ... more

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; Length and width of the rectangle are 10 units ... more

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; Length and width are both reduced by . Height ... more

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; Radius of the cylinder . Height is 3 units ... more

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. ; The height of the cone calculated by put... more

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; To obtain the quotient and the remainder, ... more

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  ; The dividend polynomial can be rewritten as , ... more

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; Let the length of the rectangle be , and Width ... more

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The binomial is a factor of given  polynomial.  ; ... more

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; The degree of the quotient polynomial is less ... more

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.  ; Divide the dividend polynomial by the ... more

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,  ; Divide the dividend polynomial by the ... more

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.  ; Divide the dividend polynomial by the ... more

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.  ; Divide the dividend polynomial by the ... more

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.  ; Divide the dividend polynomial by the ... more

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.  ; Rewrite the dividend polynomial as . Divide ... more

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.  ; Divide the dividend polynomial by the ... more

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.. ; Rewrite the dividend polynomial as . Divide ... more

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.  ; Divide the dividend polynomial by the ... more

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  ; Divide the dividend polynomial by the divisor... more

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  ; Divide the dividend polynomial by the divisor... more

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 . ; Rewrite the divisor binomial as . Divide the... more

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. ; Divide the dividend polynomial by the divisor... more

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; Determine the value of from the equation for ... more

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; The binomial is in the form of where is a real ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of where is a real ... more

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; The binomial is in the form of where is a ... more

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; The binomial is in the form of where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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. ; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of , where is a ... more

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; The binomial is in the form of where is a ... more

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; The binomial is in the form of where is a real ... more

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; The binomial is in the form of where is a real ... more

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. ; The first step in synthetic multiplication is ... more

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No. ; First, evaluate the value of from equation... more

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Yes; . ; Evaluate the value of from equation for... more

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No. ; Evaluate the value of from equation for ... more

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Yes; . ; Evaluate the value of from equation for... more

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Yes; . ; Evaluate the value of from equation for... more

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No ; Evaluate the value of from equation for ... more

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; Use division algorithm to find the other ... more

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; Use the division algorithm to find the other ... more

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  ; If the polynomial crosses the axis at then is ... more

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; If the polynomial crosses the axis at then is ... more

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polynomial in factored form is = ; using the ... more

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quotient : reminder : ; To find the quotient and ... more

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 quotient :  reminder: ; To find the quotient and ... more

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quotient : reminder : 9 ; To find the quotient and... more

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Quotient:  Reminder: ; To find the quotient and ... more

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 Quotient:  Reminder: -6 ; To find the quotient ... more

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; To evaluate the expression for , consider the ... more

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  ; To evaluate the expression for , consider the... more

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; To evaluate the expression for , consider the ... more

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; When the dividend is and the divisor is Since ... more

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; When the dividend is and the divisor is Since ... more

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; The binomial divisor is , so the divisor in ... more

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; The binomial divisor is , so the divisor in ... more

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; The binomial divisor is , so the divisor in ... more

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; The binomial divisor is , so the divisor in ... more

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; The binomial divisor is , so the divisor in ... more

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; Since the rectangle area is the product of ... more

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; Since the rectangle area is the product of ... more

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; Since the rectangle area is the product of ... more

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  ; Since the volume is the product of length, ... more

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; As the volume is the product of the length, ... more

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; Volume is the product of length, width, and ... more

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; First, divide the volume having the expression... more

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; The height of the cylinder is obtained by ... more

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; The height of the cylinder is obtained by ... more

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; The volume of the cylinder is given by . The ... more

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-412 ; Divide the polynomial by . Using the ... more

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2, -2, and -4. ; Use the synthetic division to ... more

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No rational zero exists. ; The possible rational ... more

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Zeroes, ; Since the sum of coefficient is zero, it... more

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; Assume the required polynomial be of degree 2... more

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 Positive Real ZeroesNegative Real ZeroesComplex ... more

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The required container length, width and height ... more

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Remainder theorem can be used to determine is a ... more

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As the Remainder Zero theorem used the fact that ... more

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Rational zeros are always rational numbers. But a ... more

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No changes of sign: Polynomial does not have any ... more

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Because it shows that the divisor is one of the ... more

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-6 ; The remainder theorem states that if a ... more

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-106 ; Remainder theorem states that if a ... more

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-17 ; According to remainder theorem, when a ... more

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0 ; According to remainder theorem, when a ... more

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-44791 ; According to remainder theorem, when a ... more

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255 ; According to the  remainder theorem, when a ... more

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95 ; According to the remainder theorem, when a ... more

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-1 ; According to the remainder theorem, when a ... more

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; Divide the polynomial function is by the long ... more

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; Divide the polynomial by using the long ... more

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; Divide the polynomial function by using the ... more

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-2 ; For the given polynomial, check whether is a... more

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; For the given polynomial function, check is a ... more

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-3 ; For the given polynomial function, checking ... more

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; For the given polynomial function, check ... more

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  ; Check is a factor of by synthetic division ... more

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  ; Given polynomial function is . The rational ... more

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; Given polynomial function is The Rational Zero ... more

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; Given polynomial function is The Rational Zero ... more

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; Given polynomial function is . The rational ... more

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; Given polynomial function is . The rational ... more

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; Given polynomial function is . The rational ... more

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; Given polynomial function is . The rational ... more

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; Given polynomial function is . The rational ... more

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; The rational zero theorem states that if  is a ... more

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; The rational zero theorem states that if  is a ... more

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; The rational zero theorem states that if  is a ... more

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; The rational zero theorem states that if  is a ... more

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; The rational zero theorem states that if  is a ... more

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; The rational zero theorem states that if  is a ... more

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, 2, -3 ; For the polynomial having the expression... more

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; Leading coefficient of the polynomial is 1, so ... more

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; To find possible rational roots of the ... more

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  ; To find the possible rational roots of the ... more

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; To find the possible rational roots of the ... more

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; To find the possible rational roots of the ... more

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; To find the possible rational roots of the ... more

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; To find the possible rational roots of the ... more

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; Use the rational root theorem to find the ... more

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; Use the rational root theorem to find the ... more

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Positive solutions:One.Negative solutions:Zero. ; ... more

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Positive solutions:One .Negative solutions:One. ; ... more

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Positive solutions:Two.Negative solutions:One. ; ... more

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Positive solutions:One or Three.Negative solutions... more

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Positive solutions:One. Negative solutions:One. ; ... more

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Positive solutions:Zero.Negative solutions:Three ... more

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Positive solutions:Two. Negative solutions:One. ; ... more

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Positive solutions:Two or Zero.Negative solutions:... more

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Positive solutions: Two.Negative solutions: Two... more

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Positive solutions: Two or zero.Negative solutions... more

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; Find the possible rational roots of the ... more

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; .To determine and , find all the factors of ... more

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; .To determine and , find all the factors of ... more

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. ; .To determine and , write down all the ... more

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. ; .To determine and , write the factors of the ... more

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, and 1 ; Plot the function using a graphing ... more

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-1, and ; Plot the function using a graphing ... more

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, and 2 ; Plot the function using a graphing ... more

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-3, and ; Plot the function using a graphing ... more

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; Plot the function , using a graphing utility.... more

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; From the real roots -1, 1 and 3, the factors of... more

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vertical asymptote: ;horizontal asymptote: 2.0012.... more

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vertical asymptote: horizontal asymptote: 2.9992.... more

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