Problem 41 - Elementary and Intermediate Algebra

Question

Factor expression. Factor out any GCF first. \(5 a b^{4}-5 a\)

Step-by-Step Solution

Answer

The factorized form of the expression is \(5a(b^4 - 1)\).

Step 1: Identify GCF

Start by determining the greatest common factor (GCF) of the terms in the expression. The terms are \(5ab^4\) and \(-5a\). Both terms have a common factor of \(5a\).

Step 2: Factor Out the GCF

Once we have identified the GCF, we will factor it out of the expression. This means dividing each term by the GCF and writing the expression as a product.\[\begin{align*}5ab^4 - 5a &= 5a(b^4) - 5a(1) \&= 5a(b^4 - 1).\end{align*}\]

Step 3: Confirm Factoring

Multiply the factored expression back out to confirm that it returns to the original expression. Performing the multiplication: \(5a(b^4 - 1)\) gives \(5ab^4 - 5a\), which matches the original expression.

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor, or GCF, is one of the fundamental ideas when it comes to factoring algebraic expressions. Imagine it as the biggest piece that can evenly divide all the terms in a given expression. Finding the GCF is the first step when you start factoring, as it simplifies the expression into more manageable parts.

• To identify the GCF, look for the highest number that divides the coefficients of all terms without leaving a remainder.
• Also, identify any variables that are common in all terms, with the lowest power they appear in.

In our example, with the expression \(5ab^4 - 5a\), notice that both terms have a factor of \(5a\). By pulling out this common factor, you simplify the expression and make further operations much easier.

Polynomials

At its core, a polynomial is a mathematical expression made up of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. Understanding polynomials is essential for mastery in algebra.

• Polynomials can have one or many terms, such as the simple constant \(5\) or a more complex expression like \(5ab^4 - 5a\).
• Each part of a polynomial is called a term, and terms are usually separated by plus or minus signs.

In the expression we are factoring, \(5ab^4 - 5a\), there are two terms: \(5ab^4\) and \(-5a\). Recognizing and understanding the structure helps in the process of finding common factors and organizing the expression more efficiently.

Factoring Techniques

Factoring is the art of breaking down an expression into its simpler components or factors. This is an essential skill in algebra, as it makes solving equations and simplifying expressions much easier. When factoring, especially for polynomials, the process usually starts with finding and factoring out the greatest common factor (GCF), as demonstrated in the step-by-step example. Here’s a simple approach:

• Identify and factor out the GCF from the expression, making the terms inside the parentheses less complex.
• After factoring out, multiply the factors back to ensure accuracy and ensure the original expression can be reconstructed.

In our example, after factoring out \(5a\) from \(5ab^4 - 5a\), we're left with \(5a(b^4 - 1)\). Always verify by multiplying to ensure the factorization is correct, ensuring it matches the original expression.