Problem 33 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(8 y^{3}+16 y^{2}\)

Step-by-Step Solution

Answer

8y^2(y + 2)

Step 1: Identify the Greatest Common Factor (GCF)

Examine each term in the polynomial to find the greatest common factor. The terms are: - First term: 8y^3 - Second term: 16y^2 The greatest common factor for the coefficients (8 and 16) is 8. The lowest power of y in both terms is y^2. Therefore, the GCF is 8y^2.

Step 2: Factor Out the GCF

Divide each term of the polynomial by the GCF (8y^2) and then rewrite the polynomial as a product of the GCF and the resulting polynomial: 1. \(\frac{8y^3}{8y^2} = y\) 2. \(\frac{16y^2}{8y^2} = 2\) Therefore, the factored form is: 8y^2(y + 2)

Key Concepts

Greatest Common Factor

When dealing with polynomials, one of the first steps in simplifying them is to find the Greatest Common Factor (GCF).
The GCF is the largest number and variable(s) that evenly divide all terms in the polynomial.
Identifying the GCF ensures that you can factor the polynomial correctly and simplify it to its most basic form.

• For coefficients, find the largest number that divides each term.
• For variables, use the lowest exponent that appears in all terms.

For instance, in the polynomial \(8y^3 + 16y^2\), the coefficients are 8 and 16. The GCF for these coefficients is 8.
The variable part has \(y^3\) and \(y^2\), and the lowest power that is common in both terms is \(y^2\).
Hence, the GCF of the entire polynomial is \(8y^2\).

Polynomial Factoring

Once you have identified the Greatest Common Factor (GCF), the next step is polynomial factoring.
Factoring is the process of breaking down a complex expression into a product of simpler expressions.

• First, divide each term by the GCF.
• Then, group the results inside a parenthesis.

This rewrites the original polynomial as a product of the GCF and the simplified polynomial.
For example, after finding the GCF \(8y^2\) in \(8y^3 + 16y^2\), divide each term by \(8y^2\):
\frac{8y^3}{8y^2} = y\
\frac{16y^2}{8y^2} = 2\
Hence, the original polynomial can be factored as \(8y^2(y + 2)\).

Algebra Basics

Understanding algebra basics helps in dealing with polynomials effectively.
Here are some key points to always keep in mind:

• Polynomials are expressions that include variables, coefficients, and exponents.
• Each term in a polynomial is a product of a coefficient and a variable raised to an exponent.
• You can combine like terms to simplify polynomials. Like terms are terms that contain the same variable raised to the same power.

For example, \(2x^3 + 5x - 3 + x^3 + 2\) can be simplified by combining like terms to \(3x^3 + 5x - 1\).
By mastering these basics, factoring and other algebraic operations will become more intuitive.