Problem 66 - Elementary and Intermediate Algebra

Question

Factor each polynomial by factoring out the opposite of the GCF. $$ -4 a^{2}-6 a $$

Step-by-Step Solution

Answer

The factored polynomial is \(-2a(2a + 3)\).

Step 1: Identify the GCF

First, identify the greatest common factor (GCF) of the terms in the polynomial. The terms are \( -4a^2 \) and \( -6a \). The GCF of the coefficients \( -4 \) and \( -6 \) is \( 2 \). The variable \( a \) is common, so the GCF is \( 2a \).

Step 2: Factor out the Opposite of the GCF

Now, we must factor out the opposite of the GCF from each term. The opposite of \( 2a \) is \( -2a \).Divide each term of the polynomial by \( -2a \):\[ \frac{-4a^2}{-2a} = 2a \]\[ \frac{-6a}{-2a} = 3 \]This gives us:\[ -4a^2 - 6a = -2a(2a + 3) \]

Step 3: Rewrite the Factored Expression

After factoring out \( -2a \), express the polynomial as a product:\[ -4a^2 - 6a = -2a(2a + 3) \]

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is a critical concept in algebra. It refers to the largest factor that two or more numbers or terms share. Finding the GCF is often the first step in simplifying or factoring algebraic expressions.
To find the GCF of a polynomial:

• Identify the coefficients of each term; these are the numerical parts in front of the variables.
• Examine the variables themselves; check for common variables shared by each term.
• Determine the highest power of the common variable that appears in all terms.
• The GCF will often include both a numerical part and a variable part.

As seen in our example, the terms are \(-4a^2\) and \(-6a\). The common factor among the numbers \(-4\) and \(-6\) is \(2\), while \(a\) is the common variable. Hence, the GCF of these terms is \(2a\). Sometimes, instead of factoring the GCF, you might need to factor the opposite of the GCF, which would be the negative of that factor.

Factoring Expressions

Factoring expressions is a fundamental operation in algebra that simplifies expressions or equations, making them easier to work with. When we factor an expression, we essentially "unpack" it by breaking it down into simpler components multiplied together.
Here’s how you can factor algebraic expressions:

• Begin by identifying the GCF of the terms in the expression, as this will often be the first factor you can take out.
• Once the GCF is determined, divide each term in the expression by this factor.
• The expression can then be rewritten as the product of the GCF and the remaining terms.
• In some cases, like the example above, you'll factor out the opposite of the GCF, meaning you divide by the negative of the GCF

Following these steps is typically efficient for polynomials, especially when they need to be simplified or computed in equations. In our problem, factoring out \(-2a\) from \(-4a^2 - 6a\) gives the expression \(-2a(2a + 3)\).

Algebraic Expressions

Algebraic expressions form the backbone of algebra and consist of terms formed by constants, variables, and coefficients. They can be simple or complex, depending on the number and arrangement of the terms.
A few key points to recognize an algebraic expression:

• A term in an algebraic expression can be a constant, a single variable, or a combination of both.
• Expressions can include addition, subtraction, multiplication, and division operations among variables and constants
• Terms are usually separated by plus or minus signs.
• Factors are the multiplied parts within a term.

Consider the expression \(-4a^2 - 6a\). It's made up of two terms: \(-4a^2\) and \(-6a\). Each term consists of a coefficient, a variable, and, in the case of \(-4a^2\), an exponent. Recognizing and manipulating these parts are essential skills in algebra.