Problem 4 - A Graphical Approach to College Algebra

Question

Factor the greatest common factor from each polynomial. $$28 r^{4} s^{2}+7 r^{3} s-35 r^{4} s^{3}$$

Step-by-Step Solution

Answer

The greatest common factor of the polynomial is \(7r^3s\); factored form: \(7r^3s(4rs + 1 - 5rs^2)\).

Step 1: Identify the Greatest Common Factor (GCF) for the Coefficients

First, let's analyze the coefficients of the terms in the polynomial, which are 28, 7, and -35. The greatest common factor (GCF) of these numbers is 7.

Step 2: Identify the Greatest Common Factor for the Variables

Now, examine the variable parts of each term. The terms are \(r^4s^2, r^3s,\) and \(r^4s^3\). For \(r\), the smallest power is \(r^3\), and for \(s\), it is \(s^1\). Therefore, the GCF for the variables is \(r^3s\).

Step 3: Factor Out the GCF

Multiply the GCF of the coefficients (7) by the GCF of the variables (\(r^3s\)). The overall GCF is \(7r^3s\). Now, divide each term in the polynomial by \(7r^3s\):- \(\frac{28 r^{4} s^{2}}{7r^3s} = 4rs\)- \(\frac{7 r^{3} s}{7r^3s} = 1\)- \(\frac{-35 r^{4} s^{3}}{7r^3s} = -5rs^2\)So the factored form of the polynomial is \(7r^3s(4rs + 1 - 5rs^2)\).

Step 4: Verify the Factorization

Multiply \(7r^3s\) by each term inside the parenthesis to check the original polynomial:\((7r^3s) \cdot (4rs) = 28r^4s^2\)\((7r^3s) \cdot (1) = 7r^3s\)\((7r^3s) \cdot (-5rs^2) = -35r^4s^3\)The original polynomial matches, confirming our factorization is correct.

Key Concepts

Understanding the Greatest Common Factor (GCF)

Factoring polynomials often starts with finding the **Greatest Common Factor (GCF)**. This is the largest number or expression that divides all the terms in the polynomial without leaving a remainder.
For instance, in the polynomial \(28 r^{4} s^{2} + 7 r^{3} s - 35 r^{4} s^{3}\), the coefficients are 28, 7, and -35.

• Step one is to identify the GCF of these numbers, which is the biggest number that divides into all of them.
• Here, 7 is the GCF because 7 divides 28, 7, and -35 evenly.

This approach helps simplify the polynomial by reducing the complexity of the expression, making it easier to handle and solve in subsequent steps.

Explaining Variable Factorization

**Variable factorization** is another critical part of factoring polynomials and involves finding common variables across all terms.
Each term in a polynomial can have variables raised to different powers. For \(28 r^{4} s^{2}, 7 r^{3} s\), and \(-35 r^{4} s^{3}\), we observe the variables are \(r\) and \(s\).

• To factor the variables, start by identifying the lowest power of each variable that appears in every term.
• In this example, for \(r\), the smallest power is \(r^3\), and for \(s\), it’s \(s^1\).
• This means the variable part of the GCF is \(r^3s\).

By taking the smallest powers, you ensure that each term in the polynomial can be divided by the variable GCF, simplifying the process of factoring.

Role of Coefficients in Polynomial Factorization

The **coefficients** in a polynomial are the numerical parts of the terms, and they play a vital role in factorization.
These numbers indicate how many times the variable term is counted in each part of the expression.

• In the polynomial given, the coefficients are 28, 7, and -35.
• During factorization, these numbers guide the calculation of the greatest common factor, as they must be taken into account separately from the variables.
• By focusing on 7, the GCF of the coefficients, we can simplify calculations by reducing complexity.

Effectively managing coefficients allows for streamlined factoring and helps maintain the integrity of the polynomial during simplification, ensuring all calculations remain accurate.