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Find the zeros of the function. You may want to view the graph of the function to help you identifythe real root, then use it to depress the polynomial & find the remaining roots.I

Find The Zeros Of The Function You May Want To View The Graph Of The Function To Help You Identifythe Real Root Then Use It To Depress The Polynomial Amp Find T class=

Sagot :

We must find the zeros of the following function:

[tex]f(x)=x^3-x^2-11x+15.[/tex]

1) First, we plot a graph of the function:

From the graph, we see that the function crosses the x-axis at x = 3, so x = 3 is one of the zeros of the function.

2) Because x = 3 is a zero of the function, we can factorize the function in the following way:

[tex]f(x)=x^3-x^2-11x+15=(x^2+b\cdot x+c)\cdot(x-3)\text{.}[/tex]

To find the coefficients b and c, we compute the product of the parenthesis and then we compare the different terms:

[tex]f(x)=x^3-x^2-11x+15=x^3+(b-3)\cdot x^2+(c-3b)\cdot x-3c.[/tex]

To have the same expressions at both sides of the equality we must have:

[tex]\begin{gathered} -3c=15\Rightarrow c=-\frac{15}{3}=-5, \\ b-3=-1\Rightarrow b=3-1=2. \end{gathered}[/tex]

So we have the following factorization for the function f(x):

[tex]f(x)=(x^2+2x-5)\cdot(x-3)\text{.}[/tex]

3) To find the remaining zeros, we compute the zeros of:

[tex](x^2+2x-5)\text{.}[/tex]

The zeros of this 2nd order polynomial are given by:

[tex]x=\frac{-b\pm\sqrt[]{b^2-4\cdot a\cdot c}}{2a},[/tex]

where a, b and c are the coefficients of the polynomial. In this case we have a = 1, b = 2 and c = -5. Replacing these values in the formula above, we get:

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