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Chapter 5: Problem 29

Factor completely. $$ r s^{4}+64 r s $$

Short Answer

Expert verified
The expression is factored as \( r (s^4 + 64) \).

Step by step solution

01

Identify Common Factors

First, look for common factors in all the terms. Both terms contain the factor \( r \).
02

Factor Out the Common Term

Factor \( r \) out of each term in the expression \( r s^{4} + 64 r \). This gives: \[ r (s^{4} + 64) \]
03

Recognize the Sum of Cubes

Notice that \( s^{4} + 64 \) can be written as \( (s^2)^2 + 8^2 \), which resembles the sum of squares. However, sum of squares cannot be factored further over the real numbers, so the process stops here.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Common Terms
When you factor algebraic expressions, always look for common terms first. This means finding what terms appear in each part of the expression.
Let鈥檚 take the example from the exercise: r s^{4} + 64 r Both terms have the common factor of r. To factor, we pull out this common term. This makes the expression simpler.
  • Identify the common factor: r
  • Write the expression as a product of this factor and the remaining terms. In this exercise, it becomes r (s^{4} + 64)
Factoring common terms is like finding the greatest common divisor (GCD) for numbers but for expressions.
Algebraic Expressions
Algebraic expressions contain variables, constants, and operations (addition, subtraction, multiplication, and division). In the exercise, we have r s^{4} + 64 r.
Each part of the algebraic expression is called a term. Here, r s^{4} and 64 r are both terms.
  • Variables are symbols used to represent unknown values.
  • Constants are fixed values like 64 in our example.
  • Operations link these components to form algebraic expressions.
Understanding how these parts work together helps in factoring and solving algebraic equations.
Sum of Squares
In the exercise, after factoring the common term, we get: r (s^{4} + 64).
Now, look at s^{4} + 64. This expression is a sum of squares: s^{4} + 64 = (s^{2})^{2} + 8^{2}.
The sum of squares cannot be factored further using real numbers. It's different from the difference of squares, which has a specific formula. For example: a^{2} - b^{2} = (a - b)(a + b).
However, there's no simple formula for factoring a^{2} + b^{2} over real numbers. It's important to recognize when you reach this limit in factoring.

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