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Rewrite the expression as a product of four linear factors:

(x^(2)-9x)^(2)-2(x^(2)-9x)-80
Answer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(x^{2}-9 x\right)^{2}-2\left(x^{2}-9 x\right)-80$ $\newline$ Answer:

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Complex conjugate theorem

Full solution

Q. Rewrite the expression as a product of four linear factors:

\newline

\left(x^{2}-9 x\right)^{2}-2\left(x^{2}-9 x\right)-80

\newline

Answer:

Define New Variable: Let's denote the expression inside the parentheses as a new variable to simplify the problem. Let $u = x^2 - 9x$ . The expression becomes: $\newline$ $(u)^2 - 2u - 80$
Factor Quadratic Expression: Now we have a quadratic in terms of $u$ . We can factor this quadratic expression as we would normally do with any quadratic. We are looking for two numbers that multiply to $-80$ and add up to $-2$ . These numbers are $-10$ and $8$ . So we can write: $(u - 10)(u + 8)$
Substitute Back and Simplify: Now we substitute back $x^2 - 9x$ for $u$ to get the expression in terms of $x$ : $\newline$ $(x^2 - 9x - 10)(x^2 - 9x + 8)$
Factor First Quadratic: Next, we need to factor each quadratic expression. Starting with $x^2 - 9x - 10$ , we look for two numbers that multiply to $-10$ and add up to $-9$ . These numbers are $-10$ and $1$ . So we can write: $\newline$ $(x - 10)(x + 1)$
Factor Second Quadratic: Now, we factor $x^2 - 9x + 8$ . We look for two numbers that multiply to $8$ and add up to $-9$ . These numbers are $-1$ and $-8$ . So we can write: $\newline$ $(x - 1)(x - 8)$
Write Original Expression: Finally, we write the original expression as a product of four linear factors: x - \(10)(x + $1$ )(x - $1$ )(x - $8$ )\

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