Factor the expression completely. x^(4)+8x^(2)-9 Answer:

Recognize Structure: Recognize the structure of the expression. The expression x^4 + 8x^2 - 9 resembles a quadratic in form, where x^2 is the variable instead of x. We can substitute y = x^2 to make it look like a standard quadratic equation: y^2 + 8y - 9.Factor Quadratic Expression: Factor the quadratic expression. We need to find two numbers that multiply to -9 and add up to 8. These numbers are 9 and -1. So we can write the quadratic as (y + 9)(y - 1).Substitute Back: Substitute back x^2 for y. Replace y with x^2 in the factored form to get (x^2 + 9)(x^2 - 1).Recognize Difference of Squares: Recognize that x^2 - 1 is a difference of squares. The term x^2 - 1 can be factored further since it is a difference of squares: x^2 - 1 = (x + 1)(x - 1).Write Completely Factored Expression: Write the completely factored expression. The completely factored form of the original expression is (x^2 + 9)(x + 1)(x - 1).

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Q. Factor the expression completely.

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x^{4}+8 x^{2}-9

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Answer:

Recognize Structure: Recognize the structure of the expression. $\newline$ The expression $x^4 + 8x^2 - 9$ resembles a quadratic in form, where $x^2$ is the variable instead of $x$ . We can substitute $y = x^2$ to make it look like a standard quadratic equation: $y^2 + 8y - 9$ .
Factor Quadratic Expression: Factor the quadratic expression. $\newline$ We need to find two numbers that multiply to $-9$ and add up to $8$ . These numbers are $9$ and $-1$ . So we can write the quadratic as $(y + 9)(y - 1)$ .
Substitute Back: Substitute back $x^2$ for $y$ . Replace $y$ with $x^2$ in the factored form to get $(x^2 + 9)(x^2 - 1)$ .
Recognize Difference of Squares: Recognize that $x^2 - 1$ is a difference of squares. $\newline$ The term $x^2 - 1$ can be factored further since it is a difference of squares: $x^2 - 1 = (x + 1)(x - 1)$ .
Write Completely Factored Expression: Write the completely factored expression. $\newline$ The completely factored form of the original expression is $(x^2 + 9)(x + 1)(x - 1)$ .