Factor each polynomial. See Example 7. 9(a-4)^2+30(a-4)+25

Verified step by step guidance

1. Recognize that the expression is a quadratic in terms of \((a-4)\).
2. Rewrite the expression as \(9x^2 + 30x + 25\) where \(x = (a-4)\).
3. Notice that this is a perfect square trinomial of the form \((3x + 5)^2\).
4. Factor the expression as \((3(a-4) + 5)^2\).
5. Simplify the expression to \((3(a-4) + 5)^2\).

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 understanding the polynomial's roots. Common methods include factoring by grouping, using the distributive property, and applying special formulas like the difference of squares or perfect square trinomials.

Perfect Square Trinomials

A perfect square trinomial is a specific type of polynomial that can be expressed as the square of a binomial. The general form is (a ± b)² = a² ± 2ab + b². Recognizing this pattern allows for quick factoring, as seen in the given polynomial, which can be rewritten as the square of a binomial, simplifying the factoring process.

Substitution in Polynomials

Substitution is a technique used to simplify the factoring process by replacing a complex expression with a single variable. In the given polynomial, substituting 'x' for '(a-4)' can make it easier to identify patterns and apply factoring techniques. After factoring, the original variable can be substituted back to find the final result.