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Chapter 6: Problem 54

Multiply or divide. Write each answer in lowest terms. $$ \frac{(x+4)^{3}(x-3)}{x^{2}-9} \div \frac{x^{2}+8 x+16}{x^{2}+6 x+9} $$

Short Answer

Expert verified
(x + 4) (x + 3)

Step by step solution

01

- Factor Denominators

First, factor the quadratic expressions in the denominators. Note that \(x^2 - 9 = (x-3)(x+3)\) and \(x^2 + 6x + 9 = (x + 3)^2\).
02

- Factor Numerators

Next, factor the quadratic expression in the numerator of the second fraction. Recall that \(x^2 + 8x + 16 = (x + 4)^2\).
03

- Rewrite the Expression

Rewrite the original expression using the factored forms: \[\frac{(x+4)^{3}(x-3)}{(x-3)(x+3)} \ \div \ \frac{(x+4)^{2}}{(x+3)^{2}}.\]
04

- Multiply by the Reciprocal

To divide by a fraction, multiply by its reciprocal: \[\frac{(x+4)^{3}(x-3)}{(x-3)(x+3)} \ \cdot \ \frac{(x+3)^{2}}{(x+4)^{2}}.\]
05

- Simplify the Expression

Cancel out any common factors: \[(x+4)^{3},(x+3),(x-3).\] This simplifies to \[(x+4) (x+3)\].
06

- Final Answer

Combine the remaining factors, so the final simplified expression is: \[(x+4) (x+3)\].

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Key Concepts

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

Factoring Quadratics
Factoring quadratics is a crucial skill when working with rational expressions. A quadratic expression is of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. To factor it, you should look for two numbers that multiply to \(ac\) and add up to \(b\).
For example:
  • \(x^2 - 9\) factors into \((x-3)(x+3)\). This is because \(-3\) and \(+3\) are roots of the equation \(x^2 - 9 = 0\).
  • \(x^2 + 6x + 9\) factors into \((x+3)^2\). This is because \(3\) and \(3\) are roots of the equation \(x^2 + 6x + 9 = 0\).
  • \(x^2 + 8x + 16\) factors into \((x+4)^2\). This is because \(4\) and \(4\) are roots of the equation \(x^2 + 8x + 16 = 0\).
Practicing these factorizations frequently will boost your confidence and efficiency in simplifying rational expressions.
Multiplying Fractions
Multiplying fractions is key when simplifying and manipulating rational expressions. The core rule is to multiply the numerators together and the denominators together.
Here's how you do it:
  • Multiply the numerators: If you have \( \frac{a}{b} \) and \( \frac{c}{d} \), the product is \(\frac{ac}{bd}\).
  • Use the reciprocal: When dividing fractions, you multiply by the reciprocal of the second fraction. In our problem, we convert the division \(\div\) into a multiplication by flipping the second fraction.
Applying this, you get
\[ \frac{(x+4)^{3}(x-3)}{(x-3)(x+3)} \cdot \frac{(x+3)^{2}}{(x+4)^{2}} \]
This helps to create common factors which can be cancelled out.
Simplifying Expressions
Simplifying expressions is the art of making a complex mathematical expression easier to understand and work with. Here's how:
  • Cancel common factors: Look for terms that appear both in the numerator and the denominator. In the given exercise, we have \((x+4)^2\) in the numerator and denominator which we can cancel out. Similarly, we can cancel \((x-3)\) and \((x+3)\).
  • Combine like terms: Once the common factors are canceled, multiply the remaining terms. This results in a simpler form of the expression.
Please note the step-by-step process:
  • Step 1: Cancel \((x+4)^2\), \(x-3\), and \(x+3\) to simplify to \(x + 4\) and \(x + 3\).
  • Step 2: Combine the remaining terms to reach the final answer: \((x + 4)(x + 3)\).
Always double-check that no common factors are left and your expression is in its simplest form.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{5-x}{5+x} $$

Divide. Write each answer in lowest terms. $$ \frac{(x-3)^{2}}{6 x} \div \frac{x-3}{x^{2}} $$

Multiply. Write each answer in lowest terms. $$ \frac{(a+b)^{2}}{5} \cdot \frac{30}{2(a+b)} $$

Multiply or divide as indicated. Write each answer in lowest terms. $$ \frac{2}{3} \cdot \frac{5}{6} $$

Write four equivalent forms for each rational expression. $$ -\frac{5 x-6}{x+4} $$
See all solutions

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