Problem 41 - Elementary Algebra

Question

Factor out the GCF. $$ 18 a_{2}+30 a-6 $$

Step-by-Step Solution

Answer

The factored expression is \( 6(3a^2 + 5a - 1) \).

Step 1: Identify the GCF of coefficients

Look at the coefficients of the terms: 18, 30, and 6. The greatest common factor (GCF) of these numbers is 6. This is the number that can be divided evenly into each of the coefficients.

Step 2: Write expression as a sum

Rewrite the expression as a sum of terms with the GCF emphasized: \( 18a^2 + 30a - 6 \). Each term shares the GCF of 6.

Step 3: Factor out the GCF

Divide each term by the GCF (6). This process yields: \( 6(3a^2) + 6(5a) - 6(1) \). Simplifying inside the parentheses gives: \( 6(3a^2 + 5a - 1) \). This shows the original expression factored by its GCF.

Key Concepts

Factoring Expressions

Factoring expressions involves breaking down complex algebraic expressions into simpler, multiplied components. This process is akin to unwrapping a gift to see what's inside. When we factor expressions, we aim to express them as a multiplication of simpler expressions or numbers.
Here's why factoring is important:

• It simplifies expressions and makes them easier to work with.
• It helps solve equations, especially polynomials.
• It reveals hidden structures within expressions, making mathematical expressions more readable.

For example, when we factor the expression \( 18a^2 + 30a - 6 \), we identify a common component—in this case, 6—that's present in all terms. We "factor out" this component to make the expression less complicated.

Coefficients

Coefficients are the numerical parts of the terms in an algebraic expression. In the expression \( 18a^2 + 30a - 6 \), the coefficients are the numbers 18, 30, and 6. They tell us how many times to take each of the variable terms. Recognizing coefficients is crucial to factoring expressions.

When factoring, the first step is often to look for the greatest common factor (GCF) among the coefficients. The GCF is the highest number that can evenly divide each of the coefficients. This common number is what you factor out initially to simplify the expression. This process is vital as it sets the stage for further simplification or solving of algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In essence, they are the backbone of algebra. An algebraic expression, like \( 18a^2 + 30a - 6 \), consists of different components:

• Terms: Parts of the expression separated by '+' or '-' signs. Here, \( 18a^2 \), \( 30a \), and \(-6\) are terms.
• Coefficients: Numerical parts attached to variables. 18 and 30 are coefficients here.
• Variables: Letters that represent numbers, like \( a \).

Understanding each component helps in breaking down, or factoring, expressions efficiently, making them easier to handle and solve. Mastery over algebraic expressions is key for students delving into advanced mathematics.

Polynomial Factoring

Polynomial factoring is one of the most fundamental techniques when dealing with algebraic expressions. In simple terms, it means writing a polynomial as a product of one or more simpler polynomials. For example, the polynomial \( 18a^2 + 30a - 6 \) can be factored out to \( 6(3a^2 + 5a - 1) \). By doing so, we discover a simpler form of the polynomial.

The purpose of polynomial factoring is multiple:

• To simplify polynomials for ease of handling.
• To make polynomial equations solvable.
• To identify roots or zeros of polynomial functions.

This method helps streamline complex problems into manageable operations, laying groundwork for solving equations and understanding polynomial graphs.