Problem 68 - A Graphical Approach to College Algebra

Question

Factor by any method. $$a^{3}(r+s)+b^{2}(r+s)$$

Step-by-Step Solution

Answer

\((r+s)(a^{3} + b^{2})\)

Step 1: Identify the Common Factor

Look for common factors in the terms. Both terms have a common factor of \((r + s)\).

Step 2: Factor Out the Common Factor

Write the expression as \((r+s)(a^{3} + b^{2})\) by factoring \((r+s)\) out of each term.

Key Concepts

Factoring Expressions

Factoring expressions in algebra is like breaking down a complex equation into simpler pieces. Imagine you have a big number, like 12, and you can write it as the product of smaller numbers, like 3 and 4. In algebra, when you factor an expression, you're doing something similar but with algebraic terms instead of plain numbers.
When you're faced with a complicated algebraic expression, the goal of factoring is to simplify it. This might involve finding something that all parts of the expression share, which is called a common factor. By finding and pulling out these common factors, you simplify the whole expression, making it much easier to work with. This not only helps you solve equations more easily but also helps you understand the underlying structure of the expression.

Polynomial Expressions

A polynomial expression is like a math phrase full of terms. Each term in a polynomial can be a number, a variable, or numbers and variables multiplied together. For example, in the polynomial expression like \[ a^{3}(r+s) + b^{2}(r+s), \]you can see that it has two terms. Each term is composed of different variables raised to certain powers and multiplied by coefficients, which in this case are 1 for both terms.
Polynomials can range from being very simple, like a single number, to really complex, like in the given exercise. Understanding how to manipulate these expressions is crucial in algebra because it allows you to simplify the terms and possibly rewrite them in other forms. Factoring is one common method of manipulating polynomial expressions to make them easier to work with.

Common Factor Extraction

Common factor extraction involves finding and "pulling out" elements shared by all parts of an expression. This strategy simplifies expressions and makes complex equations easier to handle. In the original exercise, \[ a^{3}(r+s) + b^{2}(r+s), \]the common factor is \( (r+s) \), since it appears in both terms.
To extract the common factor, you "factor it out" of each term. Think of this process as the reverse of the distributive property: rather than distributing \( (r+s) \) into each term, you are taking it out. After this step, the expression is reorganized as \( (r+s)(a^{3} + b^{2}) \).
This rewritten form turns a seemingly complicated expression into a more manageable one. This form is not only simpler but also versatile, making it easier to use in further algebraic manipulations or problem-solving. By understanding common factor extraction, you are equipped with a foundational tool for solving algebraic equations more efficiently.