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If
f(x)=2x^(3)+14x^(2)-32 x+36 and
x+9 is a factor of
f(x), then find all of the zeros of
f(x) algebraically.

If $f(x)=2 x^{3}+14 x^{2}-32 x+36$ and $x+9$ is a factor of $f(x)$ , then find all of the zeros of $f(x)$ algebraically.

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Prime factorization with exponents

Full solution

Q. If

f(x)=2 x^{3}+14 x^{2}-32 x+36

and

x+9

is a factor of

f(x)

, then find all of the zeros of

f(x)

algebraically.

Set Equation Equal: Since $x+9$ is a factor of $f(x)$ , we can set it equal to zero to find one of the zeros. $\newline$ $x + 9 = 0$ $\newline$ $x = -9$
Perform Polynomial Division: Now we'll perform polynomial division to divide $f(x)$ by $x+9$ and find the other factors. $\newline$ We divide $2x^3 + 14x^2 - 32x + 36$ by $x + 9$ .
Find Quadratic Zeros: After performing the division, we get a quotient of $2x^2 - 4x + 4$ . Now we need to find the zeros of this quadratic equation.

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