Problem 29 - College Algebra

Question

For the following exercises, factor the polynomial. $$ 144 b^{2}-25 c^{2} $$

Step-by-Step Solution

Answer

The polynomial \(144b^2 - 25c^2\) factors as \((12b - 5c)(12b + 5c)\).

Step 1: Identifying the Type of Polynomial

The polynomial given is \(144b^2 - 25c^2\), which is a difference of squares because both \(144b^2\) and \(25c^2\) are perfect squares.

Step 2: Express Each Term as a Square

We can express \(144b^2\) as \((12b)^2\) and \(25c^2\) as \((5c)^2\). This confirms that the polynomial is a difference of squares.

Step 3: Apply the Difference of Squares Formula

Use the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), where \(a=12b\) and \(b=5c\).

Step 4: Factor the Polynomial

Substitute \(a=12b\) and \(b=5c\) into the formula to get \((12b - 5c)(12b + 5c)\).

Key Concepts

Difference of Squares

The difference of squares is a special pattern in algebra where you have two perfect squares separated by a subtraction sign. This pattern can be represented as \(a^2 - b^2\). The key to factoring a difference of squares is recognizing this pattern and remembering its corresponding formula:

• The formula: \(a^2 - b^2 = (a - b)(a + b)\)

These two binomials are known as conjugates. The beauty of the difference of squares is that it allows us to turn a complex-looking expression into a simple product of two binomials.
In the exercise, \(144b^2 - 25c^2\) is a clear example of this pattern. You first identify the two perfect squares: \((12b)^2\) and \((5c)^2\). Once recognized, you apply the difference of squares formula, making factoring straightforward. This method is very efficient, especially when dealing with polynomials in many math problems.

Perfect Squares

Perfect squares are expressions that can be written as the square of a monomial. You're likely familiar with numbers like 1, 4, 9, and 16, which are perfect squares because they equal another integer multiplied by itself. Similarly, any expression like \((3x)^2\) or \((2y)^2\) is a perfect square because they represent a term multiplied by itself.

• Important to recognize: the square of a number or an algebraic expression
• Examples: \(x^2\), \((3y)^2\), \(4b^2\)

Identifying perfect squares is crucial when factoring because it allows you to recognize patterns, such as the difference of squares. For \(144b^2 - 25c^2\), recognizing \(144b^2\) as \((12b)^2\) and \(25c^2\) as \((5c)^2\) was an important preliminary step. It's like solving a puzzle by first identifying the corner pieces.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients, involving terms with different degrees, typically separated by addition or subtraction. They form the foundation for operations in algebra, and come into play often in factoring.

• Polynomial terms: like \(3x^2\), \(-4x\), or \(2\)
• Polynomials can be simple, like \(x - 2\), or more complex, like \(5x^2 + 3x + 1\)

Understanding these expressions is vital when trying to interpret and factor them. Recognizing specific types of polynomials can simplify these tasks.
Take the given exercise: \(144b^2 - 25c^2\). Its form — with two terms, both being perfect squares — immediately suggests certain strategies, like the difference of squares. In polynomial expressions, identifying the type of expression is often the crucial first step towards finding a viable solution method.