Factor the expression completely. x^(4)+x^(2)-20 Answer:

Recognize as quadratic in form: Recognize the expression as a quadratic in form. The given expression x^4 + x^2 - 20 can be treated as a quadratic equation by substituting x^2 for another variable, let's say y. So the expression becomes y^2 + y - 20.Factor the expression: Factor the quadratic expression. We now factor y^2 + y - 20 as if it were a standard quadratic equation. We look for two numbers that multiply to -20 and add up to 1 (the coefficient of y). These numbers are 5 and -4. So, y^2 + y - 20 factors into (y + 5)(y - 4).Substitute back x^2 for y: Substitute x^2 back for y. Now we replace y with x^2 in the factored form to get (x^2 + 5)(x^2 - 4).Recognize difference of squares: Recognize that x^2 - 4 is a difference of squares. The term x^2 - 4 can be further factored since it is a difference of squares. It factors into (x + 2)(x - 2).Write completely factored expression: Write the completely factored expression. The completely factored form of the original expression is (x^2 + 5)(x + 2)(x - 2).

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Q. Factor the expression completely.

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x^{4}+x^{2}-20

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Answer:

Recognize as quadratic in form: Recognize the expression as a quadratic in form. $\newline$ The given expression $x^4 + x^2 - 20$ can be treated as a quadratic equation by substituting $x^2$ for another variable, let's say $y$ . So the expression becomes $y^2 + y - 20$ .
Factor the expression: Factor the quadratic expression. $\newline$ We now factor $y^2 + y - 20$ as if it were a standard quadratic equation. We look for two numbers that multiply to $-20$ and add up to $1$ (the coefficient of $y$ ). These numbers are $5$ and $-4$ . $\newline$ So, $y^2 + y - 20$ factors into $(y + 5)(y - 4)$ .
Substitute back $x^2$ for $y$ : Substitute $x^2$ back for $y$ . Now we replace $y$ with $x^2$ in the factored form to get $(x^2 + 5)(x^2 - 4)$ .
Recognize difference of squares: Recognize that $x^2 - 4$ is a difference of squares. $\newline$ The term $x^2 - 4$ can be further factored since it is a difference of squares. It factors into $(x + 2)(x - 2)$ .
Write completely factored expression: Write the completely factored expression. $\newline$ The completely factored form of the original expression is $(x^2 + 5)(x + 2)(x - 2)$ .