Factor each polynomial. See Example 7. $a^4-2a^2-48$

Verified step by step guidance

1. Recognize that the polynomial \(a^4 - 2a^2 - 48\) is a quadratic in form if we let \(x = a^2\). This means the expression can be rewritten as \(x^2 - 2x - 48\).
2. Factor the quadratic expression \(x^2 - 2x - 48\) by finding two numbers that multiply to \(-48\) and add to \(-2\). These numbers will help us break down the middle term.
3. Rewrite the quadratic as \((x + m)(x + n)\) where \(m\) and \(n\) are the numbers found in the previous step. This gives the factorization in terms of \(x\).
4. Substitute back \(x = a^2\) into the factored form to get \((a^2 + m)(a^2 + n)\), expressing the factorization in terms of \(a\).
5. Check if either \(a^2 + m\) or \(a^2 + n\) can be factored further, such as by recognizing a difference of squares or other factoring techniques.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process helps simplify expressions and solve polynomial equations by breaking them down into components that are easier to work with.

Recognizing Quadratic Form

Some higher-degree polynomials can be treated as quadratic expressions by substituting a variable, such as letting u = a^2. This allows the use of quadratic factoring techniques on expressions like a^4 - 2a^2 - 48 by viewing it as u^2 - 2u - 48.

Factoring Quadratic Expressions

Factoring quadratic expressions involves finding two binomials whose product equals the quadratic. This typically requires identifying two numbers that multiply to the constant term and add to the coefficient of the middle term, enabling the polynomial to be expressed as a product of binomials.