Problem 28 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$x^{3} y^{2}-x y^{2}$$

Step-by-Step Solution

Answer

The polynomial can be factored as \(xy^2(x - 1)(x + 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look for any common factors in each term of the polynomial. The terms are \(x^3y^2\) and \(xy^2\). Both terms have a \(xy^2\) factor.

Step 2: Factoring Out the GCF

Factor out the greatest common factor, \(xy^2\), from each term. This results in the expression: \[ xy^2(x^2 - 1) \]

Step 3: Factor Remaining Polynomial

Examine the polynomial \(x^2 - 1\) to see if it can be factored further. This is a difference of squares, which can be factored as \((x - 1)(x + 1)\).

Step 4: Write Complete Factorization

Write the complete factorization of the polynomial, using both the GCF and the factored expression from the difference of squares:\[ xy^2(x - 1)(x + 1) \]

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an essential tool in simplifying expressions and tackling polynomial factorization. It refers to the largest factor that can divide each term of a polynomial without leaving a remainder.

To identify the GCF, examine all the terms of the polynomial for common variables and/or coefficients. Consider the polynomial given: \(x^3y^2 - xy^2\). Here, both terms share a common factor: \(xy^2\). This factor contains the highest power of each variable present in both terms.

By factoring out \(xy^2\), you simplify the expression and reduce it to a manageable form. Think of it as peeling off the outer layer to reveal a more straightforward problem to solve.

Difference of Squares

A difference of squares is a special type of polynomial expression. It takes the form \(a^2 - b^2\) and can be factored using the formula: \((a - b)(a + b)\). This pattern arises because multiplying two conjugate binomials always results in a difference of squares.

In our given problem, after factoring the GCF \(xy^2\), you're left with \(x^2 - 1\). This is a classic difference of squares, with \(x^2\) being \(x\) squared, and \(1\) being \(1^2\).

• \(a = x\)
• \(b = 1\)

By applying the formula, you factor \(x^2 - 1\) into \((x - 1)(x + 1)\). The difference of squares is a powerful technique, simplifying expressions and making them easier to work with, especially in polynomial factorization.

Monomial Factors

Monomial factors play a crucial role in the factorization process. A monomial is a single term, consisting of a product of numbers and variables. When dealing with polynomial factorization, it's crucial to identify any monomial factors that can be factored out first.

For the polynomial \(x^3y^2 - xy^2\), the monomial factor \(xy^2\) appears in both terms. This makes it straightforward to factor it out, simplifying the original expression. By removing the monomial factor, you simplify the polynomial, allowing you to clearly see and apply additional factoring techniques, like the difference of squares.

Recognizing and handling monomial factors early in the process ensures a smoother factorization journey, often leading to a complete and efficient solution with less complex steps.