Problem 34 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$20 r^{3} s^{3}-14 r s^{4}$$

Step-by-Step Solution

Answer

The short answer is: \(2rs^{3}(10r^{2} - 7s)\).

Step 1: Identify the Greatest Common Factor (GCF) of the numerical coefficients

First, we need to find the GCF of the numerical coefficients, which are 20 and 14. The GCF of these numbers is 2.

Step 2: Identify the GCF of the variable factors

For the variables, the given expression has the factors \(r^{3}\), \(s^{3}\), \(r\), and \(s^{4}\). The greatest common factor for the variables is the lowest power of each variable present in both terms. In this case, it is \(r\) (since \(r^1 < r^3\)) and \(s^3\) (since \(s^3 < s^4\)). So the GCF for the variables is \(r^{1}s^{3}\).

Step 3: Factor the GCF from the expression

Now that we have identified the GCF to be 2 and the variable factor to be \(rs^{3}\), we can factor them out from the given expression: \(20r^{3}s^{3} - 14rs^{4} = 2rs^{3}(10r^{2} - 7s)\)

Step 4: Check your answer

To make sure we factored the GCF correctly, we can distribute it back to both terms: \(2rs^{3}(10r^{2} - 7s) = 20r^{3}s^{3} - 14rs^{4}\) Since our factored expression expands back to the original expression, we have factored it correctly. The factored expression is: \(2rs^{3}(10r^{2} - 7s)\)

Key Concepts

greatest common factor

The **Greatest Common Factor (GCF)** is an essential concept when it comes to factoring algebraic expressions. It refers to the largest factor that divides two or more numbers, or terms, without leaving a remainder. In simpler terms, it's the biggest number that fits perfectly into both numbers. To find the GCF:

• Look at each coefficient in your terms. For example, in the expression "20" and "14," the greatest common factor is "2," because it's the largest number that divides both 20 and 14 perfectly.
• Don't forget the variables! Determine the lowest power of each variable present in the terms. In our example, we have variables "r" and "s". The smallest powers are "r" (power of 1) and "s³" (since both terms contain "s," but the smallest power is 3). Thus, the GCF for the variables is "rs³".

Identifying and understanding the GCF is like finding a common thread. It ensures the expression can be simplified or restructured accurately.

algebraic expressions

**Algebraic expressions** are the backbone of many mathematical operations, including factoring. These expressions involve numbers, variables, and operations such as addition and subtraction. Consider our earlier example: "20r³s³ - 14rs⁴" This expression consists of:

• Numerical coefficients: 20 and 14, which tell us how much of the "r" and "s" we have.
• Variables: "r" and "s," which represent unknown quantities and can be raised to different powers.
• Operations: A subtraction sign, indicating the difference between the two terms.

Understanding how to manipulate algebraic expressions allows us to transform and factor them using techniques like finding the GCF. This forms a core part of algebraic problem-solving.

polynomial factoring

**Polynomial factoring** involves breaking down a polynomial into simpler "factor" components that, when multiplied together, give the original polynomial. The aim is to simplify expressions and solve equations more easily. Let's use our expression:"20r³s³ - 14rs⁴"This is a polynomial with two terms. Factoring it involves:

• Identifying and extracting the greatest common factor, both numerical (e.g., 2) and variable (e.g., "rs³").
• Writing the original expression as a product of its GCF and a simpler polynomial: \(2rs³(10r² - 7s)\).

By factoring the polynomial, we've turned a more complex expression into a combination of factors, making it easier to work with. This is particularly useful for solving equations or simplifying expressions further. Polynomial factoring is a critical skill in algebra and lays the foundation for more advanced math challenges.