Problem 33 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 32 x y-18 x^{2} $$

Step-by-Step Solution

Answer

The GCF is \(2x\). The factored form is \(2x(16y - 9x)\).

Step 1: Identify the Terms

Look at the polynomial given: \(32xy - 18x^2\). This polynomial consists of two terms: \(32xy\) and \(-18x^2\). The task is to factor out the greatest common factor (GCF) from these terms.

Step 2: Find the GCF of the Coefficients

The coefficients of the terms are 32 and 18. To find the GCF of these two numbers, we start by listing the factors: \(32 = 2^5\) and \(18 = 2 imes 3^2\). The common factor is \(2\). So, the GCF of the coefficients is \(2\).

Step 3: Find the GCF of the Variables

Now, compare the variable parts in each term. - For the variable \(x\), the smallest power in both terms is \(x^1\). - The second term \(-18x^2\) doesn't have \(y\). Thus, the GCF of the variable parts is \(x\).

Step 4: Factor out the GCF

The GCF of the entire polynomial is the product of the GCFs from Steps 2 and 3, which is \(2x\). To factor out \(2x\) from the polynomial:\[ 32xy - 18x^2 = 2x(16y - 9x) \]This gives us the factored form.

Key Concepts

Greatest Common Factor (GCF)

Factoring polynomials often begins with identifying the **greatest common factor (GCF)**. The GCF is the largest factor that evenly divides each term of a polynomial. It is a crucial step that simplifies expressions and helps solve equations more efficiently.
To find the GCF:

• Begin by examining the coefficients of each term.
• List the factors of each coefficient.
• Identify the largest common factor.

In the given polynomial, the coefficients are 32 and 18. The factors for 32 are 1, 2, 4, 8, 16, and 32, while 18 is divisible by 1, 2, 3, 6, 9, and 18. The largest common factor between these lists is 2. This factor will be part of the GCF for the entire polynomial.

Polynomial Terms

Understanding **polynomial terms** is another essential aspect of factoring. Polynomial terms are parts of the expression that are separated by a plus or minus sign. Each term can have constants, variables, and exponents.
In the example polynomial, there are two key terms: 32xy and -18x^2.

• 32xy is the product of the coefficient 32 and the variables x and y.
• -18x^2 combines the coefficient -18 with the variable x raised to the power of 2.

Recognizing these terms and their components is important because factoring requires breaking down each term into simpler factors. Identifying the terms precisely helps find common factors effectively.

Variables in Polynomials

**Variables in polynomials** play a significant role when factoring. They represent unknown values and are part of what makes polynomials both versatile and complex.
In the given polynomial, the variables are x and y. Each variable can be raised to different powers, which affects the process of finding the GCF.

• The variable x appears as x in the first term and as x^2 in the second.
• Variable y only appears in the first term.

To determine the GCF involving variables, you look for the lowest exponent for common variables within all terms. Thus, x^{1} is chosen as part of the GCF because it is present in both terms and has the smallest exponent.

Factored Form

When a polynomial is expressed in **factored form**, it means it is written as a product of its factors. This can make solving equations much easier and can also simplify expressions significantly.
For the polynomial 32xy - 18x^2, the factored form is achieved by removing the GCF from each term. After determining the GCF of 2x, the polynomial is rewritten as:

• 32xy becomes 2x * 16y.
• -18x^2 becomes 2x * -9x.

Thus, the factored form is 2x(16y - 9x). This shows the polynomial as a product, making it simpler to work with in solving equations or further simplification. Factoring out the GCF is an essential part of simplifying polynomials.