Factor out the greatest common factor from each polynomial. See Example 1. 8k^3+24k

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the polynomial. Look at the coefficients and the variables separately.
2. The coefficients are 8 and 24. Find the GCF of these numbers, which is the largest number that divides both 8 and 24.
3. For the variables, identify the smallest power of the common variable 'k'. Here, the terms are \(k^3\) and \(k^1\), so the smallest power is \(k^1\).
4. Combine the GCF of the coefficients and the smallest power of the variable to get the overall GCF of the polynomial.
5. Factor out the GCF from each term in the polynomial, rewriting the expression as the GCF multiplied by the remaining terms.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest integer or algebraic expression that divides two or more terms without leaving a remainder. To find the GCF, identify the common factors in the coefficients and the variables of the terms. For example, in the polynomial 8k^3 + 24k, the GCF is 8k, as it is the highest factor that can be factored out from both terms.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process simplifies expressions and can make solving equations easier. In the context of the given polynomial, factoring out the GCF allows us to express the polynomial in a simpler form, which can be useful for further analysis or solving.

Polynomial Terms

Polynomial terms are the individual components of a polynomial, typically expressed in the form of coefficients and variables raised to non-negative integer powers. Each term in a polynomial is separated by addition or subtraction. Understanding the structure of polynomial terms is essential for identifying the GCF and performing the factoring process correctly.