Problem 39 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$p(8 r-3)-q(8 r-3)$$

Step-by-Step Solution

Answer

The short answer is: \(9a^2b(7ab^2-4ab+1)\).

Step 1: Find the GCF of the coefficients and the variables

First, find the GCF of the coefficients: 63, -36, and 9. The GCF of these numbers is 9. Next, find the GCF of the variables. For each variable, find the lowest exponent that appears in the terms. - For the variable \(a\), the lowest exponent is 2. - For the variable \(b\), the lowest exponent is 1. So, the GCF of the variables is \(a^2b\). Now, the overall GCF of the given expression is \(9a^2b\).

Step 2: Factor out the GCF

Now that we have found the GCF, we'll divide each term in the expression by the GCF, and then factor it out: $$ \begin{aligned} 9a^2b\left(\frac{63a^3b^3}{9a^2b}-\frac{36a^3b^2}{9a^2b}+\frac{9a^2b}{9a^2b}\right) \end{aligned} $$

Step 3: Simplify the expression

Simplify the expression inside the parentheses and find the factored form: $$ \begin{aligned} 9a^2b(7ab^2-4ab+1) \end{aligned} $$

Step 4: Check the answer

Finally, check the answer by multiplying the factored form with its factors: $$ \begin{aligned} 9a^2b(7ab^2-4ab+1) &= (9a^2b)(7ab^2) - (9a^2b)(4ab) + (9a^2b)(1)\\ &= 63a^3b^3 -36a^3b^2 + 9a^2b \end{aligned} $$ As our check returns the original expression, our factored form is correct: $$ 9a^2b(7ab^2-4ab+1) $$

Key Concepts

Greatest Common Factor

In algebra, the concept of the Greatest Common Factor (GCF) is crucial for simplifying expressions, making equations easier to solve. The GCF is the largest factor that divides two or more numbers or terms. It is used to factor out expressions efficiently. Finding the greatest common factor is about identifying both numerical and variable components of expressions that are common across all terms.

• The GCF for numbers involves finding the highest number that divides each coefficient in the expression.
• For variables, it means identifying the smallest exponent for each variable present in the terms.

Once you have detected these common factors, pull them out to simplify the expression. This method is often the first step in making complex algebraic expressions manageable, reducing them to more workable forms.

Variable Exponents

In algebraic expressions, variables often come with exponents, which indicate how many times a variable is multiplied by itself. Dealing with variable exponents is an essential skill, especially when factoring expressions or simplifying equations.

• When finding the GCF with variables, look for the variable with the lowest exponent across all terms.
• This lowest exponent determines how many times the variable can be factored out from each term.

Employing this understanding allows you to confidently factor out the variables from each term in the expression. This helps in achieving a simpler and more consolidated form where you can clearly see the underlying structure of the algebraic statement.

Factorization Process

The factorization process in algebra involves breaking down complex expressions into simpler factors. These factors, when multiplied together, reconstruct the original expression. Here’s a straightforward approach to understanding the factorization process:

• First, identify the greatest common factor (GCF) of the given expression, including both coefficients and variables.
• Then, divide each term of the expression by this GCF.
• Finally, express the original expression as a product of the GCF and the simplified expression within parentheses.

Checking your work is an important part of the process. Multiply the factors together to ensure they yield the initial expression. This verification confirms the accuracy of your factorization, ensuring that no mistakes were made during the simplification process.