Factor the expression completely. x^(4)-11x^(2)+30 Answer:

Recognize Quadratic Form: Recognize the expression as a quadratic in form. The given expression x^(4) - 11x^(2) + 30 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) - 11y + 30.Factor Quadratic Equation: Factor the quadratic equation. We need to find two numbers that multiply to 30 and add up to -11. These numbers are -5 and -6. So we can write the quadratic equation as (y - 5)(y - 6).Substitute Back x^2: Substitute back x^(2) for y. Now we replace y with x^(2) to get the factors in terms of x: (x^(2) - 5)(x^(2) - 6).Check Further Factorization: Check for further factorization. Both x^(2) - 5 and x^(2) - 6 are difference of squares, but neither 5 nor 6 are perfect squares, so they cannot be factored further using real numbers. Therefore, the expression is fully factored.

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Q. Factor the expression completely.

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x^{4}-11 x^{2}+30

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Answer:

Recognize Quadratic Form: Recognize the expression as a quadratic in form. $\newline$ The given expression $x^{4} - 11x^{2} + 30$ 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: $y^{2} - 11y + 30$ .
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ We need to find two numbers that multiply to $30$ and add up to $-11$ . These numbers are $-5$ and $-6$ . So we can write the quadratic equation as $(y - 5)(y - 6)$ .
Substitute Back $x^2$ : Substitute back $x^{2}$ for $y$ . Now we replace $y$ with $x^{2}$ to get the factors in terms of $x$ : $(x^{2} - 5)(x^{2} - 6)$ .
Check Further Factorization: Check for further factorization. $\newline$ Both $x^{2} - 5$ and $x^{2} - 6$ are difference of squares, but neither $5$ nor $6$ are perfect squares, so they cannot be factored further using real numbers. Therefore, the expression is fully factored.