Problem 245 - Elementary Algebra

Question

Factor. \(5 q^{2}-45\)

Step-by-Step Solution

Answer

The factored form is \(5(q + 3)(q - 3)\).

Step 1 - Identify the common factor

Look for the greatest common factor (GCF) that can be factored out from both terms in the expression. In this case, both terms are divisible by 5.

Step 2 - Factor out the GCF

Factor out the GCF (5) from each term: \[5q^2 - 45 = 5(q^2 - 9)\]

Step 3 - Factor the quadratic expression

Notice that the expression inside the parenthesis, \(q^2 - 9\), is a difference of squares. Use the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\). Here, \(q^2 - 9\) can be written as \((q + 3)(q - 3)\).

Step 4 - Write the final factored form

Combine the GCF with the factored form of the quadratic expression: \[5(q^2 - 9) = 5(q + 3)(q - 3)\]

Key Concepts

Common Factor

A common factor is a number or an expression that divides each term in a polynomial evenly. To find the common factor, look at each term and determine what values are shared. For example, in the expression 5q² - 45, both terms are divisible by 5.

We call this the Greatest Common Factor (GCF) because it’s the largest value that can be factored out from all terms. Here's a simple process:

• Identify the coefficient and variable parts of each term.
• Determine the highest number and any common variables that divide all terms evenly.

By factoring out the GCF, we simplify the polynomial, making further steps in factoring easier. For instance, in 5q² - 45, the GCF is 5. So, we can write:
5(q² - 9).

Difference of Squares

The difference of squares is a special type of polynomial that takes the form a² - b². This formula is very useful: a² - b² = (a + b)(a - b).

In the example of q² - 9:

• Recognize q² as a perfect square (where a = q).
• Recognize 9 as 3² (where b = 3).

Now, apply the difference of squares formula to rewrite q² - 9 as (q + 3)(q - 3).

Understanding and identifying the difference of squares helps to factorize the expression correctly and efficiently. It's essential for simplifying polynomials and finding their roots.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number or expression that can divide each term in a polynomial without leaving a remainder. Factoring out the GCF is the first step in simplifying expressions:

• List all factors of each term.
• Identify the highest common factor.

For example, in 5q² - 45, both 5q² and 45 share a factor of 5. So, the GCF is 5.

By factoring out the GCF, we get:
5(q² - 9).

This step simplifies the polynomial, making it easier to identify and apply other factoring techniques, like the difference of squares. Always begin by looking for the GCF before moving to more complex techniques. By understanding the fundamental role of the GCF, you build a strong foundation for solving polynomial equations efficiently.