Factor the expression completely. x^(4)+12x^(2)+27 Answer:

Identify Type and Patterns: Identify the type of polynomial and look for patterns. The given expression is a quadratic in form, with x^2 taking the place of x in a standard quadratic equation. We can substitute y = x^2 to make it look like a standard quadratic equation: y^2 + 12y + 27.Factor the Quadratic Expression: Factor the quadratic expression. We need to find two numbers that multiply to 27 and add up to 12. These numbers are 3 and 9. So, we can write y^2 + 12y + 27 as (y + 3)(y + 9).Substitute Back x^2: Substitute back x^2 for y. Replace y with x^2 in the factored form to get (x^2 + 3)(x^2 + 9).Check for Further Factoring: Check if further factoring is possible. The terms x^2 + 3 and x^2 + 9 cannot be factored further over the real numbers because they do not have real roots. Therefore, the expression is fully factored.

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Q. Factor the expression completely.

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x^{4}+12 x^{2}+27

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Answer:

Identify Type and Patterns: Identify the type of polynomial and look for patterns. The given expression is a quadratic in form, with $x^2$ taking the place of $x$ in a standard quadratic equation. We can substitute $y = x^2$ to make it look like a standard quadratic equation: $y^2 + 12y + 27$ .
Factor the Quadratic Expression: Factor the quadratic expression. $\newline$ We need to find two numbers that multiply to $27$ and add up to $12$ . These numbers are $3$ and $9$ . $\newline$ So, we can write $y^2 + 12y + 27$ as $(y + 3)(y + 9)$ .
Substitute Back $x^2$ : Substitute back $x^2$ for $y$ . Replace $y$ with $x^2$ in the factored form to get $(x^2 + 3)(x^2 + 9)$ .
Check for Further Factoring: Check if further factoring is possible. $\newline$ The terms $x^2 + 3$ and $x^2 + 9$ cannot be factored further over the real numbers because they do not have real roots. Therefore, the expression is fully factored.