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The polynomial p(x)=x^(3)-6x^(2)+32 has a known factor of (x-4).
Rewrite p(x) as a product of linear factors.
p(x) = ◻

The polynomial $p(x)=x^{3}-6 x^{2}+32$ has a known factor of $(x-4)$ . $\newline$ Rewrite $p(x)$ as a product of linear factors. $\newline$ $p(x) = \square$

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Transformations of absolute value functions

Full solution

Q. The polynomial

p(x)=x^{3}-6 x^{2}+32

has a known factor of

(x-4)

.

\newline

Rewrite

p(x)

as a product of linear factors.

\newline

p(x) = \square

Factor Out Known Factor: Factor out the known factor from the polynomial. $\newline$ Since we know that $(x - 4)$ is a factor of $p(x)$ , we can perform polynomial division or use synthetic division to divide $p(x)$ by $(x - 4)$ to find the other factors.
Perform Synthetic Division: Perform the synthetic division using the known factor $(x - 4)$ . We set up the synthetic division with the root of the known factor, which is $4$ , and the coefficients of $p(x)$ , which are $1$ , $-6$ , and $32$ . $\newline$ $\newline$ $4$ | $1$ $-6$ $0$ $32$ $\newline$ | $4$ $4$ $2$ $4$ $3$ $\newline$ ----------------- $\newline$ $1$ $4$ $5$ $4$ $2$ $0$ $\newline$ The result of the synthetic division gives us the coefficients of the quotient polynomial: $4$ $8$ .
Factor Quotient Polynomial: Factor the quotient polynomial. $\newline$ The quotient polynomial is a quadratic, which we can factor further if possible. We look for two numbers that multiply to $-8$ and add up to $-2$ . These numbers are $-4$ and $2$ .
Write Factored Form: Write the factored form of the quotient polynomial. $\newline$ The factored form of the quadratic is $(x - 4)(x + 2)$ . Therefore, we can write $p(x)$ as the product of its factors: $\newline$ $p(x) = (x - 4)(x - 4)(x + 2)$ .
Simplify Repeated Factor: Simplify the repeated factor. $\newline$ Since $(x - 4)$ is a factor that appears twice, we can write it as a squared term: $\newline$ $p(x) = (x - 4)^2(x + 2)$ .

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