Problem 118 - Introductory Algebra for College Students

Question

Factor completely. $$(x-6)^{2}-9 a^{2}$$

Step-by-Step Solution

Answer

The factored form of the expression \((x - 6)^{2} - 9a^{2}\) is \((x - 3a - 6)(x + 3a - 6)\).

Step 1: Identify Square Terms

The given expression \((x - 6)^{2} - 9a^{2}\) is in the form \(a^{2} - b^{2}\). Here, \(a = (x - 6)\) and \(b = 3a\).

Step 2: Apply the Difference of Squares Formula

Apply the difference of squares formula \(a^{2} - b^{2} = (a - b)(a + b)\) on the expression. Substitute \(a = (x - 6)\) and \(b = 3a\) to get \((x - 6 - 3a)(x - 6 + 3a)\)

Step 3: Simplify the Expression

Simplify the obtained expression to get, \((x - 6 - 3a)(x - 6 + 3a) = (x - 3a - 6)(x + 3a - 6)\)

Key Concepts

Difference of Squares

The difference of squares is a special algebraic formula used to factor certain types of polynomial expressions. It is represented by the equation:

• \[a^{2} - b^{2} = (a - b)(a + b)\]

This formula is useful for expressions where two perfect squares are subtracted from each other. When you recognize the form \[a^{2} - b^{2},\] you can directly apply the difference of squares formula to factor it.
In the given exercise, \[(x-6)^2 - 9a^2,\] we identify \[a = (x - 6)\] and \[b = 3a.\] Using this formula helps us simplify the expression effortlessly by transforming it into a product of two binomials. Always ensure both terms are perfect squares before employing this method.

Polynomial Expressions

Polynomial expressions consist of variables raised to different powers, combined using addition, subtraction, or multiplication. In fact, they are foundational to algebra and math problems.
A polynomial might look complex, but it can often be simplified by recognizing patterns and using factoring techniques.
The given expression, \[(x-6)^{2} - 9 a^{2},\] is a polynomial that exhibits a recognizable pattern. Here each part of the expression is either a monomial or a group of monomials. Our goal is to transform this into a factored form, utilizing algebraic formulas and identities. Recognizing polynomial patterns is crucial in easily simplifying or solving expressions. It can include connecting terms or rearranging the form of a polynomial when looking to apply specific factoring techniques.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables within them. They are formulas or rules that simplify complex expression manipulations. The difference of squares is one such identity among many in algebra.
Applying these identities makes it easier to solve equations or factor expressions.
In our exercise, having identified it as a difference of squares, we used the identity \[a^{2} - b^{2} = (a-b)(a+b).\] Knowing and using these identities allows us to approach algebra problems more systematically and reduce errors. They provide dependable shortcuts that ease the complexity of polynomial equations.
Get familiar with common algebraic identities, such as the perfect square trinomial and the difference of cubes, as they are instrumental in efficient problem-solving.