Factor out the greatest common factor from each polynomial. See Example 1. xy-5xy²

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the polynomial. The terms are \(xy\) and \(-5xy^2\). Look for common variables and coefficients in both terms.
2. Determine the GCF of the coefficients: the coefficients are 1 (implied in \(xy\)) and -5. The GCF of 1 and 5 is 1.
3. Determine the GCF of the variables: both terms have at least one \(x\) and one \(y\). The smallest power of \(x\) is \(x^1\), and the smallest power of \(y\) is \(y^1\). So, the variable part of the GCF is \(xy\).
4. Write the GCF as \(xy\) and factor it out from each term. Express the polynomial as \(xy(\text{something})\).
5. Divide each term by the GCF \(xy\) to find the terms inside the parentheses: \(\frac{xy}{xy}\) and \(\frac{-5xy^2}{xy}\). Simplify these to get the expression inside the parentheses.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest factor that divides two or more terms without leaving a remainder. In polynomials, it includes the highest power of variables and the largest numerical coefficient common to all terms. Factoring out the GCF simplifies expressions and is the first step in polynomial factorization.

Factoring Polynomials

Factoring polynomials involves rewriting the expression as a product of simpler polynomials or factors. Extracting the GCF is a fundamental factoring technique that simplifies the polynomial by removing common factors from each term, making further factorization or simplification easier.

Variables and Exponents in Terms

Each term in a polynomial may contain variables raised to powers. When factoring, identify the smallest exponent of each variable common to all terms to include in the GCF. For example, between xy and xy², the common variable factor is xy, since y has the lowest exponent 1.