Problem 34 - Algebra for College Students

Question

Factor completely. $$15 x^{4} y^{2}-45 x^{5} y^{4}$$

Step-by-Step Solution

Answer

The expression factors to \(15x^4y^2(1 - 3xy^2)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, find the greatest common factor of the terms in the expression. For the coefficients, the GCF of 15 and 45 is 15. For the variable factors, the GCF of \(x^4\) and \(x^5\) is \(x^4\), and the GCF of \(y^2\) and \(y^4\) is \(y^2\). Therefore, the GCF of the entire expression is \(15x^4y^2\).

Step 2: Factor Out the GCF

Next, divide each term in the expression by the GCF and factor it out. When we divide \(15x^4y^2\) by itself, we are left with 1. Dividing \(45x^5y^4\) by \(15x^4y^2\) leaves \(3xy^2\). Thus, when we factor out the GCF, the expression becomes: \[ 15x^4y^2(1 - 3xy^2) \]

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor, or GCF, is the largest number or variable that can evenly divide into all terms of a polynomial expression. Finding the GCF is a fundamental skill in simplifying algebraic expressions.

To determine the GCF:

• Identify the smallest power of each variable present in all terms.
• Look for the largest coefficient that divides all the numbers evenly.

In our exercise example, we need to find the GCF of the polynomial \(15x^4y^2 - 45x^5y^4\).

• The GCF of the coefficients 15 and 45 is 15 because 15 is the largest whole number that divides both numbers evenly.
• For the variable \(x\), compare \(x^4\) and \(x^5\). The lowest power is \(x^4\).
• For the variable \(y\), compare \(y^2\) and \(y^4\). The lowest power is \(y^2\).

Therefore, the GCF of the entire expression \(15x^4y^2 - 45x^5y^4\) is \(15x^4y^2\). By factoring out this GCF, you simplify the problem to its core components, making further algebraic operations much easier.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients arranged in a mathematical sentence. These expressions use addition, subtraction, multiplication, and non-negative integer exponents.

A polynomial’s structure helps determine how it can be factored. Understanding the terms and degrees is essential:

• **Terms:** These are parts of the expression separated by + or - signs. For example, \(15x^4y^2\) and \(-45x^5y^4\) are terms in our polynomial expression.
• **Degree:** This is the highest sum of the exponents in a term. In \(15x^4y^2\), the degree is 6, because \(4 + 2 = 6\).

Recognizing these components allows us to apply the correct factoring techniques and simplify expressions efficiently. In our example, the focus is not solely on splitting the terms but also recognizing their shared factors, which is key to fully factoring the polynomial.

Factoring Techniques

Factoring is the process of breaking down expressions into products of simpler components, often leading to easier solutions and simplification.

There are several methods of factoring:

• **Greatest Common Factor (GCF):** This is the first step, as seen in our example. By removing the GCF \(15x^4y^2\), we simplify the expression drastically.
• **Grouping:** This technique involves rearranging and grouping terms that have common factors. Not directly used in our problem, but crucial when expressions have more terms.
• **Difference of Squares and Other Special Cases:** Identifying these patterns can streamline the factoring process, although they are not applicable here.

In our task, once the GCF was factored out, the expression \(1 - 3xy^2\) remained. No further factoring was necessary because it is already in its simplest form.

Remember, practice with various factoring methods will increase your comfort and proficiency when handling polynomial expressions. Every expression may have multiple paths to its simplest form.