Factor the expression completely. x^(4)-3x^(2)-4 Answer:

Recognize Structure: Recognize the structure of the expression. The given expression is a quadratic in form, but with a variable to the fourth power: x^4 - 3x^2 - 4 We can treat x^2 as a single variable, let's say u, to make it look like a standard quadratic equation: u^2 - 3u - 4Factor Quadratic Expression: Factor the quadratic expression. Now, we factor u^2 - 3u - 4 as if it were a regular quadratic equation. We are looking for two numbers that multiply to -4 and add up to -3. These numbers are -4 and +1. So we have: (u - 4)(u + 1)Substitute Back: Substitute back x^2 for u. Now we replace u with x^2 to get back to the original variable: (x^2 - 4)(x^2 + 1)Factor Further: Factor further if possible. The term (x^2 + 1) is not factorable over the real numbers because it has no real roots. However, (x^2 - 4) is a difference of squares and can be factored further: (x^2 - 4) = (x - 2)(x + 2)Write Completely Factored Expression: Write the completely factored expression. Combining the factors from the previous steps, we get the completely factored form of the original expression: (x - 2)(x + 2)(x^2 + 1)

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Factor the expression completely.

x^(4)-3x^(2)-4
Answer:

Factor the expression completely. $\newline$ $x^{4}-3 x^{2}-4$ $\newline$ Answer:

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Algebra 2

Composition of linear and quadratic functions: find an equation

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Q. Factor the expression completely.

\newline

x^{4}-3 x^{2}-4

\newline

Answer:

Recognize Structure: Recognize the structure of the expression. $\newline$ The given expression is a quadratic in form, but with a variable to the fourth power: $\newline$ $x^4 - 3x^2 - 4$ $\newline$ We can treat $x^2$ as a single variable, let's say $u$ , to make it look like a standard quadratic equation: $\newline$ $u^2 - 3u - 4$
Factor Quadratic Expression: Factor the quadratic expression. $\newline$ Now, we factor $u^2 - 3u - 4$ as if it were a regular quadratic equation. We are looking for two numbers that multiply to $-4$ and add up to $-3$ . These numbers are $-4$ and $+1$ . $\newline$ So we have: $\newline$ $(u - 4)(u + 1)$
Substitute Back: Substitute back $x^2$ for $u$ . Now we replace $u$ with $x^2$ to get back to the original variable: $(x^2 - 4)(x^2 + 1)$
Factor Further: Factor further if possible. $\newline$ The term $(x^2 + 1)$ is not factorable over the real numbers because it has no real roots. However, $(x^2 - 4)$ is a difference of squares and can be factored further: $\newline$ $(x^2 - 4) = (x - 2)(x + 2)$
Write Completely Factored Expression: Write the completely factored expression. Combining the factors from the previous steps, we get the completely factored form of the original expression: x - \(2)(x + $2$ )(x^ $2$ + $1$ )\