Problem 12 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$25-49 n^{2}$$

Step-by-Step Solution

Answer

The factored form is \((5 - 7n)(5 + 7n)\).

Step 1: Identify the Difference of Squares Pattern

The given expression is \(25 - 49n^2\). This needs to be expressed in terms of squares. Recognize that both 25 and \(49n^2\) are perfect squares. The difference of squares pattern is \(a^2 - b^2 = (a - b)(a + b)\). Identify \(a^2 = 25\) and \(b^2 = 49n^2\).

Step 2: Express Each Term as a Square

Express 25 and \(49n^2\) as squares: \(25 = (5)^2\) and \(49n^2 = (7n)^2\). Therefore, we have: \((5)^2 - (7n)^2\).

Step 3: Apply the Difference of Squares Formula

According to the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), substitute \(a = 5\) and \(b = 7n\):\[(5)^2 - (7n)^2 = (5 - 7n)(5 + 7n)\].

Step 4: Write the Factored Form

The expression is now factored as \((5 - 7n)(5 + 7n)\). This is the fully factored form using the difference-of-squares pattern.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that simplifies expressions and solves equations. When you factor a polynomial, you aim to express it as a product of simpler polynomials, often revealing roots or solutions.

Think of factoring like breaking down a complex problem into smaller, more manageable parts. For example, when you have the expression \(25 - 49n^2\), factoring allows us to simplify this into \((5 - 7n)(5 + 7n)\).

Factoring also helps in simplifying fractions, solving polynomial equations, and understanding the properties of algebraic functions. Mastering this skill involves recognizing patterns and applying specific factoring techniques, such as the difference of squares.

• Recognize patterns that fit standard formulas like the difference of squares.
• Break down higher degree polynomials into products of lower degree ones.
• Simplify algebraic expressions for easier computation and understanding.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They represent quantities in mathematical form but do not have an equal sign. Algebra allows us to describe patterns and relationships precisely.

An important aspect of working with algebraic expressions is understanding how to manipulate and simplify them. Let's consider the expression from our problem, \(25 - 49n^2\):

• Identify components: Recognize the terms or parts of the expression. Here, 25 and \(49n^2\) are distinct parts.
• Patterns: Look for patterns, such as the difference of squares, that can be used to simplify the expression.
• Simplification: Use mathematical operations to transform the expression into a more useful form, like factoring.

By mastering algebraic expressions, you can solve complex problems by breaking them into smaller, more intuitive parts.

Perfect Squares

A key concept in algebra is that of perfect squares. A perfect square is a number or expression that can be written as the square of another number or expression. Recognizing these is crucial for simplifying and factoring expressions.

In the example \(25 - 49n^2\), we identify both terms as perfect squares: \(25 = (5)^2\) and \(49n^2 = (7n)^2\). This recognition allows us to apply the difference of squares formula effectively.

• Identify perfect square numbers: Numbers like 1, 4, 9, 16, 25, etc., which have integer square roots.
• Recognize algebraic perfect squares: Expressions like \(x^2\), \((3x)^2\), or \((5)^2\), which help simplify algebraic expressions.
• Utilize perfect squares in formulas: Use them in identities and formulas to factor or simplify complex expressions.

Understanding perfect squares provides the tools needed to manipulate expressions more easily and effectively in algebra.