Problem 17 - Elementary and Intermediate Algebra

Question

Factor. \(81 a^{2}-49 b^{2}\)

Step-by-Step Solution

Answer

The expression factors to \((9a + 7b)(9a - 7b)\).

Step 1: Identify the Difference of Squares

The given expression is \(81a^2 - 49b^2\). Notice that both terms are perfect squares: \(81a^2\) is \((9a)^2\) and \(49b^2\) is \((7b)^2\). This means the expression is a difference of squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(x^2 - y^2 = (x+y)(x-y)\). Here, let \(x = 9a\) and \(y = 7b\). Substitute these into the formula: \((9a + 7b)(9a - 7b)\).

Step 3: Write Down the Final Factored Form

The factored form of the expression \(81 a^{2} - 49 b^{2}\) is \((9a + 7b)(9a - 7b)\).

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that involves breaking down an expression into simpler components called "factors." When we factor a polynomial, we're finding two or more expressions that multiply together to give back the original polynomial. In terms of operations, it’s akin to reverse multiplying or "un-multiplying". This process is particularly useful for simplifying expressions and solving equations.

• A key strategy for factoring is to look for patterns or special forms within the expression, such as the difference of squares.
• By rewriting expressions in their factored form, we can easily identify solutions to equations and reveal more about the characteristics of the expressions.

Furthermore, factoring can simplify complex algebraic problems, making them much more manageable. Recognizing and applying patterns like these significantly expedites problem solving.

Perfect Squares

Perfect squares are numbers or algebraic expressions that can be represented as the square of an integer or another expression. For example, 81 is a perfect square because it can be expressed as \(9^2\); similarly, \(81a^2\) is a perfect square because it can be expressed as \((9a)^2\). Recognizing perfect squares quickly helps in factoring and identifying patterns like the difference of squares.

• Perfect squares in polynomials are easy to spot, especially when terms have even exponents, as they are typically the square of some base term.
• In our example, both \(81a^2\) and \(49b^2\) are perfect squares, making the identification of the difference of squares straightforward.

Spotting these perfect squares within an expression is often the first step in many factoring problems, especially when connected to techniques like the difference of squares.

Algebraic Expressions

Algebraic expressions are composed of variables, numbers, and operations like addition, subtraction, multiplication, and division. They are the building blocks of most algebra problems and can represent everything from simple arithmetic to complex functional relationships.

• An expression can be as simple as \(x+3\) or as complex as \(81a^2 - 49b^2\).
• It's essential to understand how to manipulate these expressions, including factoring and expanding, which is foundational in solving various mathematics problems.
• Working with algebraic expressions requires an understanding of basic algebraic rules and properties.

Learning to skillfully handle algebraic expressions allows students to solve problems more efficiently and prepares them for more advanced topics in mathematics.