Problem 5 - Algebra 2

Question

Factor each polynomial. \(4 x y^{2}-16 x\)

Step-by-Step Solution

Answer

The factored form is \(4x(y^{2} - 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to look at the coefficients and variables in the expression \(4xy^{2} - 16x\) to determine the greatest common factor (GCF). Both terms have a factor of \(4x\).

Step 2: Factor Out the GCF

Once we identify \(4x\) as the GCF, we need to factor it out from both terms in the expression. This means we'll divide each term in \(4xy^{2} - 16x\) by \(4x\).

Step 3: Simplify Each Term

Divide \(4xy^{2}\) by \(4x\) to get \(y^{2}\). Then, divide \(-16x\) by \(4x\) to get \(-4\).

Step 4: Write the Factored Expression

After factoring out the GCF and simplifying, rewrite the expression. The original polynomial \(4xy^{2} - 16x\) is now written as \(4x(y^{2} - 4)\).

Key Concepts

Greatest Common Factor (GCF)

Understanding the Greatest Common Factor (GCF) is crucial when factoring polynomials. Simply put, the GCF is the largest factor that divides all the terms in a polynomial without a remainder.
Let's break it down:

• Identify all the factors of each term in the polynomial.
• For example, in the expression \(4xy^2 - 16x\), the factors of \(4xy^2\) are \(1, 2, 4, x, y, xy, xy^2, 2xy, 4xy, 2xy^2, 4xy^2\).
• Meanwhile, the factors of \(-16x\) are \(1, 2, 4, 8, 16, x, 2x, 4x, 8x, 16x\).
• Identify the largest common factors between these terms, which are \(4\) and \(x\).

In our given example, the GCF is \(4x\). Finding this common factor simplifies the entire factoring process, making it easier to handle complex algebraic expressions.

Algebra 2 Problems Made Simple

Algebra 2 frequently challenges students with more complex problems that can often be simplified by factoring. Factoring is a key technique that breaks down polynomials into simpler terms. This process often involves:

• Identifying patterns, such as recognizing common factors.


• Understanding the structure of polynomial expressions, such as terms and coefficients.


• Practicing division of each term by the GCF to ensure each remaining term is simplified correctly.

In the exercise with \(4xy^2 - 16x\), once the GCF \(4x\) is factored out, the expression simplifies to \(4x(y^2 - 4)\). Recognizing and practicing these steps allows students to solve complex Algebra 2 problems with greater ease and confidence.

Polynomial Expressions and Their Components

Polynomial expressions are equations that consist of variables (letters) and coefficients (numbers). These can be in multiple terms, connected by addition or subtraction. Here's what you need to know:

• Each term in a polynomial expression is a product of coefficients and variables raised to a power.


• The degree of the polynomial is determined by the highest power of the variable present.


• Simplifying these expressions often involves combining like terms or factoring out the GCF, as shown in our exercise.

Considering the expression \(4xy^2 - 16x\), it has two terms: \(4xy^2\) and \(-16x\). By factoring out the greatest common factor, which is \(4x\), the polynomial expression is rewritten in a simpler form. This simplification is essential in solving polynomial equations efficiently and confidently.