Factor out the greatest common factor from each polynomial. See Example 1. 8k³+24k

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the polynomial \$8k^3 + 24k\(. Start by finding the GCF of the coefficients 8 and 24, and then find the lowest power of the variable \)k$ common to both terms.
2. The GCF of the coefficients 8 and 24 is 8, and the lowest power of \(k\) common to both terms is \(k^1\), so the overall GCF is \$8k$.
3. Rewrite each term as a product of the GCF and another factor: express \$8k^3\( as \(8k \times k^2\) and \)24k$ as \(8k \times 3\).
4. Factor out the GCF \$8k\( from the polynomial, which gives \)8k(k^2 + 3)$.
5. Verify the factorization by distributing \$8k$ back through the parentheses to ensure it matches the original polynomial.

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 greatest integer that divides all coefficients. Identifying the GCF is the first step in factoring expressions efficiently.

Factoring Polynomials

Factoring polynomials involves rewriting the expression as a product of simpler polynomials or factors. Factoring out the GCF simplifies the polynomial and makes further factoring or solving easier. It is a fundamental skill in algebra for simplifying expressions and solving equations.

Exponent Rules for Variables

When factoring variables with exponents, use the rule that the GCF includes the variable raised to the lowest exponent present in all terms. For example, between k^3 and k, the GCF includes k^1 because 1 is the smaller exponent. This ensures the factor is common to all terms.