Problem 65 - Elementary and Intermediate Algebra

Question

Factor. $$ a^{4}-144 b^{2} $$

Step-by-Step Solution

Answer

The expression factors into \((a^2 - 12b)(a^2 + 12b)\).

Step 1: Recognize the Expression Type

Identify the expression to see if it matches a recognizable formula. Notice that this expression, \( a^4 - 144b^2 \), is a difference of squares, because \( 144b^2 \) is a perfect square, as is \( a^4 \), since \( a^4 = (a^2)^2 \) and \( 144b^2 = (12b)^2 \).

Step 2: Apply Difference of Squares Formula

Recall that the difference of squares formula is \( x^2 - y^2 = (x-y)(x+y) \). Apply this formula where \( x = a^2 \) and \( y = 12b \), to factor the expression as \( (a^2 - 12b)(a^2 + 12b) \).

Step 3: Check Each Factor for Further Factorization

Inspect each factor to see if it can be further factored. In this case, \( a^2 - 12b \) and \( a^2 + 12b \) cannot be factored further using real numbers as they do not represent a difference or sum of squares or any other recognizable factorable expression.

Key Concepts

Difference of Squares

A common occurrence in algebra is the need to factor expressions. One of the most useful techniques in this context is recognizing the "Difference of Squares". It's a specific algebraic situation where two squared terms are subtracted from one another, forming an expression like \( x^2 - y^2 \). This can be factored into \((x-y)(x+y)\).

This fact is derived because when you expand \((x-y)(x+y)\), it results in \(x^2 - y^2\), which shows how multiplication and subtraction work together to form the squared terms. To effectively use this method, always verify that both components are indeed perfect squares, and the operation between them is a subtraction.

Identifying and applying the "Difference of Squares" formula can greatly simplify expressions, making calculations easier and solutions more apparent.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can represent real-world problems or purely mathematical challenges.

Recognizing patterns within these expressions can lead to simplifying or solving them, often through factoring. Factoring involves rewriting the expression in a more compact form, typically as a product of terms, which can be particularly useful for solving equations.

In the expression in our exercise, \(a^4 - 144b^2\), recognizing that it follows the pattern of the "Difference of Squares" requires understanding that both \(a^4\) and \(144b^2\) are perfect squares. This helps in setting up the expression such that you can then apply mathematical principles to break it down into simpler parts.

Perfect Squares

A perfect square is a number or an expression that can be expressed as the square of a whole number or a polynomial. Identifying perfect squares in algebraic expressions is critical for utilizing specific factoring techniques, like the "Difference of Squares".

For instance, in our expression, \(a^4\) is a perfect square because it is equivalent to \((a^2)^2\), and \(144b^2\) is also a perfect square, as it equals \((12b)^2\).

Knowing that an expression is a perfect square involves understanding its structure. It can lead to simplifications, enabling straightforward applications of factoring formulas, which play a crucial role in solving algebraic equations. Recognizing perfect squares is a foundational skill that enhances problem-solving abilities and mathematical understanding.