Problem 62 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 25 x^{2}-16 y^{2} $$

Step-by-Step Solution

Answer

The expression factors to \((5x - 4y)(5x + 4y)\).

Step 1: Identify the Form

The expression \( 25x^2 - 16y^2 \) is a difference of squares. It matches the form \( a^2 - b^2 \).

Step 2: Recognize the Squares

In the expression \( 25x^2 - 16y^2 \), \( 25x^2 \) is \( (5x)^2 \) and \( 16y^2 \) is \( (4y)^2 \).

Step 3: Apply the Difference of Squares Formula

The formula for factoring a difference of squares is \( a^2 - b^2 = (a-b)(a+b) \). Here, \( a = 5x \) and \( b = 4y \).

Step 4: Write the Factored Form

Using the values for \( a \) and \( b \), the expression \( 25x^2 - 16y^2 \) factors to \( (5x - 4y)(5x + 4y) \).

Key Concepts

Factoring Expressions

Factoring expressions is a crucial skill in algebra that involves rewriting a mathematical expression as a product of its factors. This process simplifies the expression, making it easier to work with.

Here’s what you need to know:

• Start by identifying the type of factoring needed: common factors, difference of squares, trinomials, etc.
• For each type, there are specific patterns and formulas to apply.
• The goal is to express the original complicated expression as a multiplication of simpler expressions.

Factoring requires careful observation and practice. It’s like breaking down a big problem into smaller, manageable parts.

Difference of Squares

The difference of squares is a specific type of factoring technique.

When you see an expression of the form \(a^2 - b^2\), it is a difference of squares. It has a particular formula for factoring: \(a^2 - b^2 = (a-b)(a+b)\). This formula is derived from the algebraic identity that describes the relationship between squares and their factors.

Here's a step-by-step guide:

• Ensure your expression matches the form \(a^2 - b^2\).
• Recognize and rewrite each square term: for example, \(25x^2\) becomes \((5x)^2\) and \(16y^2\) becomes \((4y)^2\).
• Apply the formula: for \(5x\) and \(4y\), the expression becomes \((5x - 4y)(5x + 4y)\).

Using this technique, you can quickly factor expressions that fit this specific pattern.

Polynomials

Polynomials are algebraic expressions that include terms with variables raised to whole number powers. They form the foundation for many areas of algebra.

Key characteristics of polynomials:

• Each term in a polynomial consists of a coefficient (a number) and a variable raised to a power.
• They can have one or more terms, such as in binomials (two terms) or trinomials (three terms).
• Polynomials can be added, subtracted, multiplied, and factored.

Polynomials are everywhere in algebra, and understanding how to manipulate them—through operations like factoring—is crucial for solving more complex equations.