Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 4 u^{2}-2 u v $$

Step-by-Step Solution

Answer

The polynomial is factored as \(2u(2u - v)\).

Step 1: Identify Common Factors

First, we need to look at each term in the polynomial to identify any common factors. Our polynomial is \(4u^2 - 2uv\). Notice that both terms have a factor of 2 and they both have a variable \(u\). We can factor the common terms out of the polynomial.

Step 2: Factor Out the Common Factor

After identifying the common factors as \(2u\), factor \(2u\) out of each term in the polynomial.\[4u^2 - 2uv = 2u(2u - v)\]

Step 3: Rewrite the Expression

Once the common factor is factored out, the expression inside the parentheses \(2u - v\) is what remains after dividing each term by \(2u\). So, the final factored form of the polynomial is:\[2u(2u - v)\]

Key Concepts

Common Factor

When working with polynomials, identifying a common factor is often the first step in simplifying or factoring the expression. A common factor is a factor that is shared by all terms in the polynomial. In the polynomial expression \(4u^2 - 2uv\), each term can be divided by a common factor. To identify it, we look for both numerical and variable components that are common:

• Numerical Factor: Both 4 and 2 are divisible by 2.
• Variable Factor: Both terms include the variable \(u\).

Thus, the common factor of the polynomial is \(2u\). Recognizing this is crucial as it simplifies the further factoring process.
Factoring out the common factor reduces complexity and prepares the polynomial for more advanced operations. Always begin by identifying such factors for an efficient factorization process.

Polynomial Expression

A polynomial expression consists of variables and coefficients, combined using addition, subtraction, and multiplication. In our example, we have the polynomial \(4u^2 - 2uv\):

• The expression is made up of two terms: \(4u^2\) and \(-2uv\).
• Each term in a polynomial is a product of a coefficient (like 4 and -2) and variables raised to whole number exponents (like \(u^2\) and \(uv\)).

Understanding the structure of a polynomial expression is important because it guides us in operations like adding, subtracting, and especially factoring. Polynomials can have numerous terms and complexities; however, the goal in simplifying them usually revolves around finding factors or roots. Recognizing these basic components makes it easier to handle more complex polynomials later on.

Factoring Process

The factoring process breaks down a polynomial into simpler components that can be multiplied to obtain the original expression. It's a bit like unwrapping a gift to see what's inside! Here’s a quick guide to factoring polynomials like \(4u^2 - 2uv\):

Identifying and Factoring Out the Common Factor

Start by identifying any common factors in each term. From our example, we noted that \(2u\) was common:

• Take \(2u\) out of both terms: \(4u^2 - 2uv = 2u(2u - v)\).
• This involves dividing each original term by \(2u\), simplifying the equation.

The Result

The expression inside the parenthesis, \(2u - v\), is what's left after factoring out \(2u\). The whole factored expression becomes \(2u(2u - v)\), a much simpler form of the original polynomial.
Factoring is not just a mathematical step but a way to uncover simplified expressions, solve equations, and understand polynomial functions better. By repeating the process, one can become adept at recognizing patterns and breaking down even more complex polynomials.