Problem 33 - Algebra for College Students

Question

Factor completely. $$12 x^{3} y^{4}-39 x^{4} y^{3}$$

Step-by-Step Solution

Answer

The expression is factored as \(3x^3y^3(4y - 13x)\).

Step 1: Identify the Greatest Common Factor (GCF)

Start by examining both terms in the expression: 1. The coefficients are 12 and 39. The GCF of 12 and 39 is 3.2. For the variable part, we have powers of \(x\): \(x^3\) and \(x^4\), so the GCF is \(x^3\).3. For the \(y\) terms, we have \(y^4\) and \(y^3\), so the GCF here is \(y^3\).Therefore, the overall GCF is \(3x^3y^3\).

Step 2: Factor Out the GCF

Divide each term of the expression by the GCF \(3x^3y^3\) to factor it out:1. Divide the first term: \(\frac{12x^3y^4}{3x^3y^3} = 4y\).2. Divide the second term: \(\frac{39x^4y^3}{3x^3y^3} = 13x\).The factored expression becomes:\[3x^3y^3(4y - 13x)\]

Step 3: Rearrange the Final Factored Expression

Write the final factored form, where the common factor is outside the parentheses, and the simplified expression is inside:\[3x^3y^3(4y - 13x)\]

Key Concepts

Greatest Common Factor (GCF)

In the realm of algebra, the greatest common factor (GCF) plays a pivotal role in simplifying expressions, especially when we are dealing with polynomials. The GCF is the largest factor that two or more terms have in common.

This process involves:

• Identifying the greatest number that divides each of the coefficients without a remainder. For instance, the numbers 12 and 39 have a GCF of 3.
• Looking at the variables, where the GCF is determined by the smallest power of each variable present in all terms. In our problem, for variable \(x\), both terms contain \(x^3\) and \(x^4\), so the smallest power is \(x^3\). Similarly, for variable \(y\), the terms \(y^4\) and \(y^3\) suggest that the GCF is \(y^3\).

By understanding the GCF, we can significantly reduce a polynomial expression to make it more manageable.

Polynomial Factoring

Factoring polynomials is a method used to simplify expressions by rewriting them as a product of their simplest factors.

Here's how it generally works:

• The first step involves identifying the GCF, which we pull out from the expression. This gives us a simpler expression inside the parentheses.
• Divide each term of the polynomial by the GCF. This rearrangement separates the expression into a common factor and a remaining polynomial.

Take, for example, our expression \(12x^3y^4 - 39x^4y^3\). We found that \(3x^3y^3\) is the GCF.

When we factor this out, the expression becomes \(3x^3y^3(4y - 13x)\). This step makes the expression easier to work with and further simplifies potential manipulation in algebraic operations.

Variable Powers

Variable powers are critical in determining the structure of a polynomial during the factoring process.

Let's take a closer look at the impact of variable powers on factoring:

• The expressions often contain variables raised to different powers. In our example, \(x\) and \(y\) are raised to the powers of 3 and 4.
• When determining the GCF for a collection of terms, we select the smallest exponents of each variable present across all terms as part of the GCF. Thus, for variables with powers, we choose the lowest power.

Understanding variable powers is essential because

they indicate how many times a variable is multiplied by itself, which directly influences how you simplify and factor polynomials. Recognizing and working with these powers can significantly streamline solving polynomial equations.