Factor the expression completely. x^(4)+5x^(2)-24 Answer:

Recognize Quadratic Form: Recognize the expression as a quadratic in form. The given expression x^(4) + 5x^(2) - 24 can be treated as a quadratic equation if we let y = x^(2). This gives us y^(2) + 5y - 24.Factor Quadratic Expression: Factor the quadratic expression. We need to find two numbers that multiply to -24 and add up to 5. These numbers are 8 and -3. So we can write y^(2) + 5y - 24 as (y + 8)(y - 3).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) + 8)(x^(2) - 3).Check Further Factoring: Check if the factors can be factored further. The term x^(2) + 8 cannot be factored further over the real numbers because it does not correspond to the difference of squares and does not have real roots. The term x^(2) - 3 is already in its simplest form as the difference of squares. Therefore, the expression is fully factored.

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Q. Factor the expression completely.

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x^{4}+5 x^{2}-24

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Answer:

Recognize Quadratic Form: Recognize the expression as a quadratic in form. $\newline$ The given expression $x^{4} + 5x^{2} - 24$ can be treated as a quadratic equation if we let $y = x^{2}$ . This gives us $y^{2} + 5y - 24$ .
Factor Quadratic Expression: Factor the quadratic expression. $\newline$ We need to find two numbers that multiply to $-24$ and add up to $5$ . These numbers are $8$ and $-3$ . So we can write $y^{2} + 5y - 24$ as $(y + 8)(y - 3)$ .
Substitute Back $x^2$ : Substitute back $x^{2}$ for $y$ . $\newline$ Now we replace $y$ with $x^{2}$ to get the factors in terms of $x$ : $(x^{2} + 8)(x^{2} - 3)$ .
Check Further Factoring: Check if the factors can be factored further. $\newline$ The term $x^{2} + 8$ cannot be factored further over the real numbers because it does not correspond to the difference of squares and does not have real roots. The term $x^{2} - 3$ is already in its simplest form as the difference of squares. Therefore, the expression is fully factored.