Problem 35 - Elementary and Intermediate Algebra

Question

Factor each polynomial by factoring out the opposite of the GCF. $$ -18 a^{2} b+12 a b^{2} $$

Step-by-Step Solution

Answer

The factored expression is \(-6ab(3a - 2b)\).

Step 1: Identify the GCF

Determine the greatest common factor (GCF) of the terms in the polynomial. For the terms \(-18a^2b\) and \(12ab^2\), the GCF is \(6ab\).

Step 2: Determine the Opposite of the GCF

The problem asks us to factor out the opposite of the GCF. The opposite of our GCF \(6ab\) is \(-6ab\).

Step 3: Factor Out the Opposite of the GCF

We factor \(-6ab\) out of each term in the polynomial:\(-18a^2b + 12ab^2 = -6ab(3a - 2b)\).

Step 4: Verify the Factored Form

Distribute \(-6ab\) back into the expression \((3a - 2b)\) to verify it produces the original polynomial: \(-6ab imes 3a = -18a^2band -6ab imes (-2b) = 12ab^2\). Thus, the factored form is correct.

Key Concepts

Greatest Common Factor

When dealing with polynomials, a crucial step is identifying the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms in the polynomial without leaving any remainder. This helps in simplifying polynomials and is a foundation for various factoring techniques.
To find the GCF in a polynomial, examine each term. Here’s how:

• Identify any common numerical factors. For instance, between the numbers \(-18\) and \(12\), the GCF is \(6\).
• Determine common variables, considering the lowest power of each appearing variable. In \(-18a^{2}b\) and \(12ab^{2}\), the variable \(a\) appears in common with the lowest power of \(a\) being \(a\). The same method applies for \(b\), where the lowest power present is \(b\).

Combine these factors (numerical and variable) to form the GCF. In our problem, combining these gives us \(6ab\) as the GCF.

Factoring by Opposites

Factoring by opposites is a nuanced and essential technique, particularly when an exercise explicitly asks for it. It involves separating out the negative of the GCF from the polynomial.
This process requires flipping the sign of the GCF. For example, if your GCF is \(6ab\), then its opposite is \(-6ab\).
The next task is to factor this opposite of the GCF out of the polynomial. For the given polynomial, \(-18a^{2}b + 12ab^{2}\), factoring out \(-6ab\) results in \(-6ab(3a - 2b)\). Notice how the sign change affects the terms inside the parenthesis. Factoring by opposites is especially useful when aiming to simplify expressions or solving equations that benefit from rearrangements like this. It’s important to remember that when dealing with the opposite, you’re essentially looking to bring uniformity to the signs within your polynomial.

Polynomial Verification

To ensure that the factoring process is accurate, polynomial verification is an important final step. It involves using distribution to expand the factored polynomial and confirm that it matches the original expression.
Let’s apply this to the polynomial we factored: \(-6ab(3a - 2b)\). We distribute \(-6ab\) to each term inside the parenthesis, which gives us:

• \(-6ab \times 3a = -18a^2b\)
• \(-6ab \times (-2b) = 12ab^2\)

When these calculations are performed, the results \(-18a^2b + 12ab^2\) reconstruct the original polynomial. This verification step confirms that the process of factoring by opposites was done correctly, a necessary measure to avoid errors and ascertain the accuracy of the math problem-solving process.