Problem 29 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The polynomial is factored as \((12b + 5c)(12b - 5c)\).

Step 1: Identify the Form

Notice that the polynomial is in the form \(a^2 - b^2\), where \(a = 12b\) and \(b = 5c\). This is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a+b)(a-b)\).

Step 2: Express Each Term as a Square

Write \(144b^2\) as \((12b)^2\) and \(25c^2\) as \((5c)^2\). This will allow us to directly apply the difference of squares formula.

Step 3: Apply the Difference of Squares Formula

Substitute \(a = 12b\) and \(b = 5c\) into the difference of squares formula to factor the expression: \((12b + 5c)(12b - 5c)\).

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic pattern where you have the subtraction of one squared term from another. It's expressed in the form \(a^2 - b^2\). Recognizing this pattern is crucial because it allows you to factor the expression easily using the simple formula \(a^2 - b^2 = (a + b)(a - b)\).

This formula shows that the expression can be rewritten as the product of two binomials. To apply this formula effectively, each term must be a perfect square. In the exercise example, \(144b^2\) is written as \((12b)^2\) and \(25c^2\) as \((5c)^2\), fitting perfectly into the difference of squares pattern. This makes factoring straightforward and quick, as seen when broken into its components, resulting in \((12b + 5c)(12b - 5c)\).

Remember, identifying the difference of squares is often easier if you become familiar with recognizing perfect squares and practice rewriting numbers and variables into squared terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the backbone of algebra. Understanding how to manipulate them is key to solving problems. An expression like \(144b^2 - 25c^2\) consists of multiple parts:

• Terms: These are the parts separated by plus or minus signs. Here, \(144b^2\) and \(-25c^2\) are the terms.
• Coefficients: The numbers in front of the variables, such as 144 and -25, suggest how the variables are scaled.
• Variables: Symbols like \(b\) and \(c\) represent numbers that can vary.
• Exponents: The small numbers, like the 2 in \(b^2\), show how many times the variable is multiplied by itself.

Algebraic expressions can often be simplified or factored. In this exercise, the expression is simplified using the difference of squares pattern. Recognizing and manipulating these parts appropriately allows for quick and accurate factoring, essential for solving algebraic equations.

Quadratic Expressions

A quadratic expression is any expression that includes a variable raised to the second power. It typically follows the general form \(ax^2 + bx + c\), but can appear in other forms like \(a^2 - b^2\) as well. In this context, quadratic expressions are strictly those with the highest degree of 2, meaning the greatest exponent on any of the variables is 2.

For the exercise \(144b^2 - 25c^2\), it might not look like the standard quadratic form, but it indeed is a special kind of quadratic expression, namely a difference of squares—it has no linear \(b\) term. Understanding this helps in applying correct factoring techniques. Quadratics can be solved in various ways: factoring, completing the square, or using the quadratic formula. For this one, we use the factorization approach because it is set up as a difference of squares.

Thus, knowing how to identify quadratic expressions and apply the appropriate strategies for solving them is an invaluable part of mastering algebra.