Factor the expression completely. x^(4)+7x^(2)-18 Answer:

Recognize Structure: Recognize the structure of the expression. The given expression x^4 + 7x^2 - 18 is 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 + 7y - 18.Factor Quadratic Expression: Factor the quadratic expression. We need to find two numbers that multiply to -18 and add up to 7. These numbers are 9 and -2 because 9 * (-2) = -18 and 9 + (-2) = 7. So, we can write the quadratic as (y + 9)(y - 2).Substitute Back: Substitute back x^2 for y. Replace y with x^2 in the factored form to get (x^2 + 9)(x^2 - 2).Check Further Factorization: Check for further factorization. The term x^2 + 9 cannot be factored further over the real numbers because it has no real roots. The term x^2 - 2 can be factored as a difference of squares into (x - √2)(x + √2).Write Final Form: Write the final factored form. The fully factored form of the expression is (x^2 + 9)(x - √2)(x + √2).

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Q. Factor the expression completely.

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x^{4}+7 x^{2}-18

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Answer:

Recognize Structure: Recognize the structure of the expression. $\newline$ The given expression $x^4 + 7x^2 - 18$ is 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: $y^2 + 7y - 18$ .
Factor Quadratic Expression: Factor the quadratic expression. $\newline$ We need to find two numbers that multiply to $-18$ and add up to $7$ . These numbers are $9$ and $-2$ because $9 \times (-2) = -18$ and $9 + (-2) = 7$ . $\newline$ So, we can write the quadratic as $(y + 9)(y - 2)$ .
Substitute Back: Substitute back $x^2$ for $y$ . Replace $y$ with $x^2$ in the factored form to get $(x^2 + 9)(x^2 - 2)$ .
Check Further Factorization: Check for further factorization. The term $x^2 + 9$ cannot be factored further over the real numbers because it has no real roots. The term $x^2 - 2$ can be factored as a difference of squares into $(x - \sqrt{2})(x + \sqrt{2})$ .
Write Final Form: Write the final factored form. $\newline$ The fully factored form of the expression is $(x^2 + 9)(x - \sqrt{2})(x + \sqrt{2})$ .