Problem 26 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$u^{3} v^{4}-u^{6} v$$

Step-by-Step Solution

Answer

The factored form is \(u^{3} v (v^{3} - u^{3})\).

Step 1: Identify Common Factors

First, look at both terms of the polynomial \(u^{3} v^{4}\) and \(u^{6} v\). Recognize the greatest common factor (GCF) from both terms. The GCF for the variable \(u\) is \(u^{3}\) and for the variable \(v\) is \(v\). So, the overall GCF is \(u^{3}v\).

Step 2: Factor Out the GCF

Now, factor the GCF \(u^{3}v\) out of the polynomial. This involves dividing each term by \(u^{3}v\). For the first term, \(\frac{u^{3} v^{4}}{u^{3} v} = v^{3}\). For the second term, \(\frac{u^{6} v}{u^{3} v} = u^{3}\).

Step 3: Write the Factored Form

Combine the results of step 2 to express the polynomial in its factored form. This will give you: \(u^{3} v (v^{3} - u^{3})\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is a fundamental concept in algebra that deals with finding the largest factor shared by two or more terms. This is crucial when simplifying algebraic expressions, especially when factoring polynomials.

When we're presented with a polynomial, our first objective is to identify the GCF of all the terms involved. This step is essential because factoring out the GCF simplifies the expression, making it easier to handle. For example, in the expression provided, \(u^3 v^4 - u^6 v\), each term contains factors of both \(u\) and \(v\). The GCF here is \(u^3v\), as it is the highest power of \(u\) and \(v\) that can divide all terms without leaving a remainder.

Understanding the GCF helps in breaking down complex algebraic expressions and is the first step to finding their factored form.

Algebraic Expressions

Algebraic expressions represent a fundamental element in mathematics that consist of variables, constants, and operations. They are the building blocks of polynomials, equations, and formulas.

In the given problem, the algebraic expression consists of two terms: \(u^3 v^4\) and \(u^6 v\). Each term in an algebraic expression can have multiple parts such as coefficients, variables, and their respective exponents. The key to managing these expressions is understanding how these components work together:

• Variables like \(u\) and \(v\) represent unknown quantities.
• Exponents indicate how many times a variable is multiplied by itself.

Mastering these elements helps in simplifying, solving, and ultimately factoring expressions, which is a critical skill in algebra.

Factored Form

The factored form of a polynomial is a way of expressing the polynomial as a product of its factors. This is often the goal when working with polynomials, as it can simplify further operations like solving equations or finding roots.

After identifying and factoring out the GCF from each term, as we did with \(u^3v\) from the initial polynomial \(u^3v^4 - u^6v\), we arrived at \(u^3v(v^3 - u^3)\). This expression is now in its factored form. The process involves writing each term as a product of common factors, significantly breaking down the polynomial.

Factored forms are not only simpler but also reveal more about the polynomial, such as potential roots and patterns within the expression. Understanding factored forms is essential for solving more complex algebraic problems.