Problem 22 - Elementary and Intermediate Algebra

Question

Factor each polynomial. $$ 63 x^{3} y^{2}+81 x^{2} y^{4} $$

Step-by-Step Solution

Answer

The factored form is \(9x^2y^2(7x + 9y^2)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to find the greatest common factor of all the terms in the polynomial. Look at the coefficients and the variable parts separately. For the coefficients, the GCF of 63 and 81 is 9. For the variable part, notice the smallest power of each variable between the terms: for \(x\) it's \(x^2\), and for \(y\) it's \(y^2\). Therefore, the GCF of the entire polynomial is \(9x^2y^2\).

Step 2: Factor Out the GCF

Now that we have identified the GCF \(9x^2y^2\), we will factor it out of each term in the polynomial. Divide each term by \(9x^2y^2\): 1. \(63x^3y^2 \div 9x^2y^2 = 7x\)2. \(81x^2y^4 \div 9x^2y^2 = 9y^2\)Thus, the original polynomial can be rewritten as \(9x^2y^2(7x + 9y^2)\).

Step 3: Recheck Each Factored Term

To ensure we have factored correctly, distribute the \(9x^2y^2\) back into the parenthesis: - \(9x^2y^2 imes 7x = 63x^3y^2\)- \(9x^2y^2 imes 9y^2 = 81x^2y^4\)Both expanded terms are the original polynomial terms, confirming the factorization is correct.

Key Concepts

Greatest Common Factor

When factoring polynomials, one of the first things to look for is the Greatest Common Factor (GCF). This is the largest factor that divides all terms in the polynomial without leaving a remainder. Identifying the GCF is crucial because it simplifies the polynomial by reducing its terms.
To find the GCF:

• Focus on the numerical coefficients first. For example, if you have coefficients like 63 and 81, you find their GCF by determining the largest number that divides both, which is 9 in this case.
• Consider the variable part separately. For each variable present in the terms, take the smallest power. In our exercise, for the variable \( x \), the smaller power between \( x^3 \) and \( x^2 \) is \( x^2 \). Similarly, for \( y \), it’s \( y^2 \) between \( y^2 \) and \( y^4 \).

Thus, combining the GCF of the coefficients (9) and the variables (\( x^2y^2 \)), we find that the GCF of the polynomial \( 63x^3y^2 + 81x^2y^4 \) is \( 9x^2y^2 \).

Variables in Polynomials

In polynomials, variables can take on many forms and powers. Understanding how to work with them is essential for factoring and simplifying expressions.
Polynomials can consist of:

• Single-variable terms like \( x^3 \) or \( y^4 \).
• Multivariable terms like \( x^2y^4 \), where each variable is raised to a specific power.

To factor polynomials involving variables, consider:

• The coefficients of the variables. They can give clues for identifying common factors.
• The powers of the variables. Always look for the smallest power of each variable across all terms to include in the GCF.

In our example, the terms \( 63x^3y^2 \) and \( 81x^2y^4 \) have variables where you choose the least degree for each variable to include in the GCF, resulting in including \( x^2 \) and \( y^2 \).

Distributive Property

The distributive property is a fundamental principle in algebra that helps with simplifying expressions and equations. It states that a single term multiplied by a sum is the same as multiplying each addend separately and then adding the results.
Mathematically, it's expressed as:\[ a(b + c) = ab + ac \]In the context of factoring polynomials, after factoring out the GCF from a polynomial, you apply the distributive property to check your work. For instance:

• You take the GCF, which is \( 9x^2y^2 \), and multiply it by each term inside the parenthesis \((7x + 9y^2)\).

This multiplication should return the original polynomial \( 63x^3y^2 + 81x^2y^4 \). If it does, you know the factorization is correct. The distributive property is thus used to ensure that no step in the factorization process is missed or incorrect, confirming accuracy.