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

Verified step by step guidance

1. Identify the structure of the polynomial: \(a^4 - 2a^2 - 48\) resembles a quadratic form, where \(a^4\) is like \(x^2\), \(-2a^2\) is like \(x\), and \(-48\) is the constant term.
2. Make a substitution to simplify the expression: Let \(u = a^2\). Then the polynomial becomes \(u^2 - 2u - 48\).
3. Factor the quadratic expression \(u^2 - 2u - 48\). Look for two numbers that multiply to \(-48\) and add to \(-2\). These numbers are \(-8\) and \$6$.
4. Rewrite the quadratic as \((u - 8)(u + 6)\) using the numbers found in the previous step.
5. Substitute back \(u = a^2\) into the factored form to get \((a^2 - 8)(a^2 + 6)\). This is the factorization of the original polynomial.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions, solving equations, and analyzing polynomial behavior. Common methods include factoring out the greatest common factor, using special product formulas, and applying techniques like grouping or the quadratic formula.

Quadratic Form

The expression a^4 - 2a^2 - 48 can be transformed into a quadratic form by substituting a^2 with a new variable, say 'x'. This allows us to rewrite the polynomial as x^2 - 2x - 48, which is easier to factor. Recognizing and manipulating polynomials into quadratic forms is a key strategy in polynomial factorization.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, at least one of the factors must be zero. This principle is crucial when solving polynomial equations after factoring, as it allows us to set each factor equal to zero to find the roots of the polynomial. Understanding this property is fundamental for solving equations derived from factored polynomials.