A Graphical Approach to Algebra & Trigonometry (7th Edition)

Chapter 4

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A polynomial function, of odd degree can have two... more

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The viewing window is too small to see the correct... more

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4 ; The function has 3 extrema hence the first ... more

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5 ; has 4 extrema. So number of zeroes of first ... more

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: maximum: minimum ; From the graph it is clear ... more

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: maximum: minimum ; The points of extrema are ... more

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: absolute maximumNo absolute minimum ; Of the two... more

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No absolute maximumNo absolute minimum ; On the ... more

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Values of local maximum: Value of local minimum: ... more

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Local maximum values: Local minimum values: No ... more

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End behavior:  ; As seen in the graph, on the ... more

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End behavior: , ; From the graph it is seen the ... more

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End behavior: , ; From the graph it is seen the ... more

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End behavior: , ; From the graph it is seen the ... more

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End behavior: , ; From the graph it is seen the ... more

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End behavior: , ; From the graph it is seen the ... more

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End behavior: , ; From the graph it is seen the ... more

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End behavior: , ; From the graph it is seen the ... more

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End behavior: , ; From the graph it is seen the ... more

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End behavior: , ; From the graph it is seen the ... more

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Input the function in any graph utility platform ... more

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Input the function in any graph utility platform ... more

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As the power is increases the graph comes more ... more

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As the power is increases the graph comes more ... more

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For the interval [] , the values of the function ... more

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For the interval [] , the values of the function ... more

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The behavior of the function is as follows:As , ... more

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The behavior of the function is as follows:As , ... more

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The behavior of the function is as follows:As , ... more

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The behavior of the function is as follows:As , ... more

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Number of x-intercepts is 1.The function cuts the ... more

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Number of x-intercepts is 1.The function cuts the ... more

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Number of x-intercepts is 1.The function cuts the ... more

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0 ; Input the function in any graph utility ... more

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Input the function in any graph utility platform ... more

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Input the function in any graph utility platform ... more

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Input the function in any graph utility platform ... more

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Since the function has negative leading ... more

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Since the leading coefficient of the given ... more

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Input the function in any graph utility platform ... more

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No.of points at which the function touches or cuts... more

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No.of real zeroes is the no.of points at which the... more

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ends in opposite direction and graph lies in first... more

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odd degree function as ends are in opposite ... more

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; number of real roots is determined by the ... more

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both the graphs are odd degree as ends heads in ... more

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odd degree polynomial as ends are opposite.It cuts... more

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; The number of cuts made on x-axis determines ... more

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; The points where the graph meets the x-axis are... more

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the point at which the graph attains a minimum ... more

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 maxima lies in first quadrant at infinity and ... more

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x-coordinate>0 and y-coordinate>0 i.e. first ... more

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Plotting the graph by hand:Plot the graph of .... more

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Shift horizontally the graph of towards left by 1... more

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Shift the graph of horizontally towards right by ... more

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shift the graph of horizontally towards right by ... more

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[-5,5], [10,20] ; Find the y intercept by ... more

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[-5,5], [10,-30] ; Find the y intercept by ... more

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[-15,15],[20,-150] ; Find the y intercept by ... more

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[-10,10], [-125,20] ; Find the y intercept by ... more

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[-5,5], [-10,90] ; Find the y intercept by ... more

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[-5,5], [-10,200] ; Find the y intercept by ... more

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; Cancel out the common factors. ; more

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; Cancel out the common factors. ; more

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; Cancel out the common factors. ; more

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; The Intermediate Value theorem assures that ... more

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; The Intermediate Value theorem assures that ... more

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; The Intermediate Value theorem assures that ... more

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; The Intermediate Value theorem assures that ... more

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; The Intermediate Value theorem assures that ... more

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; The Intermediate Value theorem assures that ... more

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; The Intermediate Value theorem assures that ... more

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; The Intermediate Value theorem assures that ... more

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; First, find ; In the next step, find ; It can ... more

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; First, find ; In the next step, find ; It can ... more

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There exists atleast one real zero between 2 and 2... more

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After plotting the line joining the points (3,-4) ... more

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; First, observe that the highest power in the ... more

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; Start the division by observing the highest ... more

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  ; Observe the highest power of the dividend. For... more

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; Start the division by observing the highest ... more

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; Start the long division by observing the ... more

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  ; Start the long division by observing the ... more

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; Start the long division by observing the ... more

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  ; Start the long division by observing the ... more

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; Start the long division with observing the ... more

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; Start the long division by observing the ... more

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  ; Start the long division by observing the ... more

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; Start the long division by carefully observing ... more

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0 ; Write the value of the given K on the left ... more

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0 ; Write the given value of K and list the ... more

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-25 ; Write the given value of K and then list the... more

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-2 ; Write the given value of K and then list the ... more

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; Process of using synthetic division to evaluate... more

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; Process of using synthetic division to evaluate... more

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; Process of using synthetic division to evaluate... more

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; Process of using synthetic division to evaluate... more

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; Process of using synthetic division to evaluate... more

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; Process of using synthetic division to evaluate... more

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; Process of using synthetic division to evaluate... more

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; Process of using synthetic division to evaluate... more

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Yes. 2 is a zero of given polynomial. ; Process of... more

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-1 is not the zero of the given polynomial. ; ... more

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is not the zero of the given polynomial. ; ... more

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is the zero of the given polynomial. ; Process of... more

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is the zero of the given polynomial. ; Process of... more

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is the zero of the given polynomial. ; Process of... more

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is not the zero of the given polynomial. ; ... more

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is not the zero of the given polynomial. ; ... more

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is the zero of the given polynomial. ; Process of... more

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is the zero of the given polynomial. ; Process of... more

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; Solution of the equation includes all the ... more

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; Zeros of the equation includes all the values ... more

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; since are the roots of the equation. the ... more

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; divide the given polynomial by ; for finding ... more

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; on analyzing the graph given in the question it... more

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; the required reminder can be found by dividing ... more

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; divide the given polynomial with (since is a ... more

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; find the factor of the given polynomial by ... more

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; find the other factors of the given cubic ... more

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;         The polynomial has a zero for number -... more

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; divide the given cubic equation with (since is... more

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; for finding the other roots of the equation ,... more

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; divide the given equation with (this is done ... more

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; divide the given equation by (this is done in ... more

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; divide the given cubic equation with (this done... more

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; divide the given cubic equation by (this is ... more

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; divide the give equation by (since is a root ... more

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; divide the given cubic equation with (since is... more

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; divide the given equation by (since is a root ... more

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; divide the cubic equation with (since is a ... more

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; divide the given equation by (since is a root ... more

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; divide the given equation with (since is a ... more

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; Synthetic division method is used to factor . ... more

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; Synthetic division method is used to factor . ... more

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; Synthetic division method is used to factor . ... more

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; Synthetic division method is used to factor . ... more

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; A polynomial can be divided by the other ... more

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; A polynomial can be divided by the other ... more

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; A polynomial can be divided by the other ... more

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; A polynomial can be divided by the other ... more

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; A polynomial can be divided by the other ... more

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; A polynomial can be divided by the other ... more

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. ; A polynomial can be divided by the another ... more

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; A polynomial can be divided by the other ... more

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. ; A polynomial can be divided by the other ... more

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; A polynomial can be divided by the other ... more

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; A polynomial can be divided by the another ... more

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; A polynomial can be divided by the other ... more

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The end behavior is as tends to increasing or ... more

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Maximum number of extreme: 2Maximum number of ... more

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The end behavior is as tends to increasing or ... more

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The graph sheet is taken and , are drawn.  ... more

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Extreme points are approximately (2,7,-0.5) and (... more

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The x-intercepts are (-1,0),(2.5,0)(3,0)and y-... more

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Quotient is ; Divide by x and proceed with that ... more

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-35 ; by using Synthetic division , divide \(x^4-... more

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Yes, -1 is a zero of P(x). ; Use synthetic ... more

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Linear factors of P(x)= are (x+4),(x+6) and (x-7... more

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; The graph of Y1 and it has 3 zero's and the ... more

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for function y1, behavior of its graph at x=zero ... more

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; We need to find the polynomial function with ... more

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; Given its is an polynomial function of degree 3... more

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; We need to find the polynomial function with ... more

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; We need to find the polynomial function with ... more

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; We need to find the polynomial function with ... more

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; We need to find the polynomial function with ... more

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; Given we need to find the polynomial of degree ... more

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; Given zero's of the polynomial functions as -3... more

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; Given zero's of the polynomial functions as 1,-... more

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; Given zero's of the polynomial functions as 1,-... more

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; Given zero's of the polynomial functions as 2,5... more

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; Given zero's of the polynomial functions as 4 ... more

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The final polynomial P(x) is ; The given roots of ... more

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The other two roots of the polynomial are -1-i and... more

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The remaining zeros of the polynomial given are 2-... more

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The other roots of the equation are 1+2i and 1-2i... more

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The other zeros are ; Since two zeros are given to... more

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The other zeros of the equation are -3i,   ; Since... more

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The other zeros of the polynomial are ; Since one ... more

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The polynomial P(x) here will be ; Here we have ... more

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The polynomial function P(x) here will be ; Since ... more

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The Polynomial function P(x) here will be ; We ... more

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The polynomial function here is ; The given roots ... more

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The polynomial function P(x) here will be ; The ... more

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The polynomial function here will be ; Since 4 ... more

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The polynomial function here will be ; The given ... more

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The polynomial function P(x) here will be ; The ... more

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The polynomial function here will be ; The given ... more

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The polynomial function P(x) here will be ; The ... more

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The Polynomial function here will be ; The given ... more

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The polynomial function here will be ; The given ... more

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The required graph is ; Firstly, Calculate the x ... more

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Obtain the points on curve, plug in the random ... more

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Obtain the points on curve, plug in the random ... more

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Obtain the points on curve, plug in the random ... more

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Obtain the points on curve , plug in the random ... more

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Obtain the points on curve, plug in the random ... more

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Obtain the points on curve, plug in the random ... more

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Obtain the points on curve, plug in the random ... more

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Obtain the points on curve, plug in the random ... more

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Obtain the points on curve, plug in the random ... more

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P(x)= (x+1)(x)(x-2) ; x- intercepts x1= -1x2=0x3=... more

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f(x)=(x+2)(x-1)(x-2) ; x-intercepts x1= -2x2=1x3=... more

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; x- interceptsx1=-1 x2=-2x3=1Using formula f(x)=... more

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; x-intercepts x1=-1x2=-2x3=1x4=2Using formula:-f... more

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; x- interceptsx1= -4x2=-1x3=2x4=1Using formulaf(... more

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; x - interceptsx1= - 1x2= 0x3= 2Using formulaf(x... more

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P(x)=(x+2)(x+1)(x+2)(x-3)  ; P(x)=x4+2x3-7x2-20x-... more

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;     P(1)=152P(-1)=0 ; (x+1) is a factor of x5+... more

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If n is odd , it will have at least 1 real zero ... more

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A polynomial of degree 4 will have maximum  of 4 ... more

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; The rational zero theorem represents that if ... more

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; The rational zero theorem represents that if ... more

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; The rational zero theorem represents that if ... more

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; The rational zero theorem represents that if ... more

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, , ; It is known that multiplying a polynomial by... more

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, ; It is known that multiplying a polynomial by a... more

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, ; It is known that multiplying a polynomial by a... more

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, ; It is known that multiplying a polynomial by a... more

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; Let is a rational zero written in the lowest ... more

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; Let is a rational zero written in the lowest ... more

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; Let is a rational zero written in the lowest ... more

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; Let is a rational zero written in the lowest ... more

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; Here, is a zero of the polynomial of . Since ... more

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; Here, is a zero of the polynomial of . Since ... more

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; Here, is a zero of the polynomial of . Since ... more

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; Here, is a zero of the polynomial of . Since ... more

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; Here, is a zero of the polynomial of . Since ... more

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; Here, is a zero of the polynomial of . Since ... more

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Possible real zeros:- 0 or 2 positive real zeros, ... more

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Possible real zeros:- 1 positive real zero, 0 or 2... more

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Possible real zeros are 1 positive real zero and 1... more

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Possible real zeros are 1 positive real zero and 3... more

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Possible real zeros- 2 or 0 positive real zeros, 3... more

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Possible real zeros:  4 or 2 or 0 positive real ... more

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Here, has all the coefficients real and a leading... more

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Here, has all the coefficients real and a leading... more

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Here, has all the coefficients real and a leading... more

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Here, has all the coefficients real and a leading... more

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Here, has all the coefficients real and a leading... more

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Synthetically divide the function by .   All the ... more

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Synthetically divide the function by .   All the ... more

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Synthetically divide the function by .   All the ... more

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; According to the graph there are zeros of the ... more

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; According to the graph there are zeros of the ... more

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The solution set is  {0,-5,5} ;  Taking the common... more

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The solution set is {0, -2, 3} ; Take out the ... more

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The real solution set is {2, -2} ; Subtracting 4 ... more

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The real solution set {1, -1} ; Subtracting 6 on ... more

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The real solution sets{0, -3, 6} ; Take out the ... more

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The solution sets {0, -1, 1} ; substitute, and  ... more

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The real solution sets {0,1} ; Adding on both ... more

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The solution sets {0, -1, 1} ; Subtracting on ... more

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The solution sets {0, , } ; Subtracting on both ... more

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The solution sets {0, , 3} ; Subtracting on both ... more

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The solution sets {, -2, 2} ; Using Distributive ... more

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The solution set {, -3, 3} ; Using Distributive ... more

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The complex solution sets are {, } ; Its least ... more

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The complex sets are {, } ; Taking the common ... more

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is the solution set of the given equation. ; To ... more

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is the solution set of the given equation. ; To ... more

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; To solve the given equation, follow the steps:... more

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; To solve the given equation, follow the steps:... more

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To solve the given equation, follow the steps:... more

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To solve the given equation, follow the steps:... more

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To solve the given equation, follow the steps:... more

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To solve the given equation, follow the steps:... more

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To solve the given equation, follow the steps:Take... more

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To solve the given equation, follow the steps:Take... more

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To solve the given equation, first take out ... more

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To solve the given equation, first take out ... more

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To solve , follow the steps:Take common factors ... more

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To solve , follow the steps:Take common factors ... more

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To solve , follow the steps:Take out the common ... more

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To solve , follow the steps:Take out the common ... more

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To solve , follow the steps:Take out common ... more

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To solve , follow the steps:Take out the common ... more

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To solve the given equation , follow the steps:... more

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To solve the given equation , follow the steps:... more

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Solving the equation by equating ; Solving  The ... more

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Solving the equation by equating .   ; Solving  ... more

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; Graph the polynomial equation that is . From ... more

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; Graph the polynomial equation that is . From ... more

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; Graph the polynomial equation that is . From ... more

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; Graph the polynomial equation that is . From ... more

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; The given equation can be written as , so plot ... more

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; The given equation can be written as , so plot ... more

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; From given equation, express -1 as the perfect ... more

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; From given equation, express -4 as the perfect ... more

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; Add 1 to the both sides of the equation and ... more

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; Add 8 to the both sides of the equation and ... more

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; Subtract 27 from both sides of the equation and... more

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; Subtract 64 from both sides of the equation and... more

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; Subtract 16 from both sides of the equation and... more

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; Subtract 81 from both sides of the equation and... more

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; Add 64 to the both sides of the equation and ... more

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; Add 27 to the both sides of the equation and ... more

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; On both sides of the equation, take the square ... more

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; Apply square root on both sides of the equation... more

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; Substitute in the function and simplify as ... more

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; For the equation By the Calculator,The roots ... more

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; The base of rectangle=x-(-x)=2xand the height ... more

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; The volume of tank=The volume of half sphere+... more

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; Since is the zero of the polynomial P(x) then... more

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; Congugate theorem says if is the zero of a ... more

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; Since is zero of the polynomial P(x) which ... more

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; Since is the zero of the polynomial P(x) then ... more

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; Using Rational Zeros theorem ,If is a rational ... more

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; Range of any function measn the output get ... more

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 End behaviour :The given polynomial is even ... more

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2 ; If a polynomial is of degree  is given then ... more

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Using intermediate value theorem  for the ... more

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; -7 ; If P(x) is a polynomial divided by then ... more

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; Use the long division method, to divide the ... more

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; Use the long division method, to divide the ... more

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; To find , use the synthetic division method. ... more

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; To find , use the synthetic division method. ... more

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; To find , use the synthetic division method. ... more

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; To find , use the synthetic division method. ... more

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; To find the factors from the zeros of the ... more

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; Let the polynomial function be .Using the given... more

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; Let the polynomial function be .Using the given... more

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; Let the polynomial function be .Using the given... more

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As , -1 is not a zero of the polynomial . ; ... more

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By using long division method, divide the ... more

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; If is the zero of polynomial function then the... more

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; Polynomial function has real coefficients and ... more

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, , and ; Polynomial function has real ... more

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Possible rational zeros are  Rational zeros are ; ... more

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Positive real zero is 1 and negative real zeros is... more

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Synthetically divide the function by . All the ... more

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Polynomial function has 3 real solutions and 3 is ... more

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; is a factor of the polynomial and is used to ... more

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; The function was observed to be the product of... more

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; The zeroes obtained are . The fractions are ... more

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; The zeroes of the given polynomial are ... more

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Since both the ends of the function, go in the ... more

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Since the ends of the graph of the function, go ... more

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Since the left end of the graph of the function, ... more

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No solution ; The number of real solutions that a ... more

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   ; The solution set that satisfies the ... more

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; The solution set satisfies the inequality is ... more

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; The solution set which satisfies the inequality... more

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Since the number of intercepts is equal to the ... more

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The maximum number of zeroes of a polynomial is ... more

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A polynomial of degree can have upto local ... more

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The intercept is the point at which . Therefore ... more

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Since the coordinate is positive and the graph of... more

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A polynomial with an even degree and even leading ... more

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A polynomial with nonreal complex zeroes with have... more

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2 ; To plot the graph of the expression , ... more

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; Since at the x-axis, the y-coordinate becomes ... more

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; Factor is in the form so arrange divisor 5 and ... more

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  ; Using graphing calculator, plot the graph of ... more

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; Group the first two terms and the last two ... more

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; Now, since cube has become a cuboid after ... more

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; Use the 4-digit telephone number say 123-2 and ... more

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; In the polynomial deduced from 4-digit ... more

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In the graphing calculator, enter the function of ... more

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Local minima: Local maxima: Domain: Range: ... more

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Mark the intercepts from the simplified equation. ... more

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; We have given that zero's of the required ... more

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 =0 or ; Given equation .We need to factorize the ... more

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; Given that height of the rectangular box is ,... more

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; For fitting the given values in a cubic ... more

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; We found that the sight distance for each ... more

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