Factor the expression completely. x^(4)+4x^(2)-5 Answer:

Recognize Structure: Recognize the structure of the expression. The given expression x^4 + 4x^2 - 5 resembles a quadratic in form, where x^2 is the variable. We can substitute y = x^2 to make it look like a quadratic equation: y^2 + 4y - 5.Factor Quadratic Expression: Factor the quadratic expression. We now factor y^2 + 4y - 5 as if it were a regular quadratic equation. We look for two numbers that multiply to -5 and add up to 4. These numbers are 5 and -1. So, y^2 + 4y - 5 factors into (y + 5)(y - 1).Substitute Back: Substitute back x^2 for y. Now we replace y with x^2 in the factored form to get (x^2 + 5)(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. It factors into (x + 1)(x - 1).Write Completely Factored Expression: Write the completely factored expression. The completely factored form of the original expression is (x^2 + 5)(x + 1)(x - 1).

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Q. Factor the expression completely.

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x^{4}+4 x^{2}-5

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Answer:

Recognize Structure: Recognize the structure of the expression. $\newline$ The given expression $x^4 + 4x^2 - 5$ resembles a quadratic in form, where $x^2$ is the variable. We can substitute $y = x^2$ to make it look like a quadratic equation: $y^2 + 4y - 5$ .
Factor Quadratic Expression: Factor the quadratic expression. $\newline$ We now factor $y^2 + 4y - 5$ as if it were a regular quadratic equation. We look for two numbers that multiply to $-5$ and add up to $4$ . These numbers are $5$ and $-1$ . $\newline$ So, $y^2 + 4y - 5$ factors into $(y + 5)(y - 1)$ .
Substitute Back: Substitute back $x^2$ for $y$ . Now we replace $y$ with $x^2$ in the factored form to get $(x^2 + 5)(x^2 - 1)$ .
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. It factors into $(x + 1)(x - 1)$ .
Write Completely Factored Expression: Write the completely factored expression. $\newline$ The completely factored form of the original expression is $(x^2 + 5)(x + 1)(x - 1)$ .