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Chapter 9: Problem 37

Add or subtract. Simplify where possible. \(\frac{5 x}{x^{2}-x-6}-\frac{4}{x^{2}+4 x+4}\)

Short Answer

Expert verified
The simplified form of the given expression is \(\frac{5x^2 + 6x + 12}{(x-3)(x+2)^{2}}\).

Step by step solution

01

Simplify the denominators

First, simplify the denominators. They can be factored as follows: \(x^{2}-x-6 = (x-3)(x+2)\) and \(x^{2}+4x+4 = (x + 2)^{2}\). So the expression becomes \(\frac{5x}{(x-3)(x+2)} - \frac{4}{(x+2)^{2}}\).
02

Find a common denominator

The common denominator for these two fractions would be a combination of the two denominators: \((x-3)(x+2)^{2}\). Align the fractions to the same denominator: \(\frac{5x(x+2)}{(x-3)(x+2)^{2}} - \frac{4(x-3)}{(x-3)(x+2)^{2}}\).
03

Simplify the expression

Combine the two fractions: \(\frac{5x(x+2) - 4(x-3)}{(x-3)(x+2)^{2}} = \frac{5x^2 + 10x - 4x + 12}{(x-3)(x+2)^{2}} = \frac{5x^2 + 6x + 12}{(x-3)(x+2)^{2}}\). This is the simplified expression.

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a crucial skill when dealing with rational expressions. It involves breaking down a polynomial into simpler "factorable" pieces that can be multiplied to form the original polynomial. This is similar to factoring numbers, but with variables.
Let's see how it works in our example.
  • Consider the polynomial denominator \(x^2 - x - 6\). To factor it, think about two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\), so the factored form is \((x - 3)(x + 2)\).
  • The second polynomial \(x^2 + 4x + 4\) can also be factored. Here, we look for two numbers that multiply to \(4\) and add to \(4\). These are \(2\) and \(2\), giving us \((x + 2)(x + 2)\) or \((x + 2)^2\).
Effectively factoring polynomials simplifies the expressions and allows us to perform operations like addition or subtraction by finding common denominators.
Common Denominator
A common denominator is necessary to add or subtract fractions. It ensures that fractions are expressed with the same denominator, enabling straightforward arithmetic. Finding a common denominator involves aligning the denominators of two or more fractions to be the same.
  • In our expression, the denominators are \((x-3)(x+2)\) and \((x+2)^{2}\). To find a common denominator, we combine these two parts to cover all variable terms: \((x-3)(x+2)^2\).
  • Once we determine the common denominator, the next step is to adjust each fraction to have this denominator. This may involve multiplying the numerator and denominator of each fraction by the necessary factors.
Aligning fractions to a common denominator is essential for simplifying operations involving polynomial fractions, ensuring that they can be summarized into a single fraction.
Simplifying Expressions
Simplifying expressions is the final step in dealing with rational expressions. After you've aligned the fractions to a common denominator, combining them and reducing them to their simplest form is crucial for clarity.
  • Combine the fractions by aligning their numerators over the common denominator. In our exercise, this means merging \(5x(x+2)\) and \(-4(x-3)\).
  • Expand the expressions: \(5x(x+2)\) becomes \(5x^2 + 10x\), and \(-4(x-3)\) becomes \(-4x + 12\).
  • Combine these into one expression: \(5x^2 + 10x - 4x + 12\), which simplifies to \(5x^2 + 6x + 12\).
This final expression represents the simplest form of your original operation. Each of these steps is vital to ensure that you're working with the expression in its simplest and most understandable form, making it easier to deal with any future operations or evaluations.

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Most popular questions from this chapter

Solve each equation. Check your answer. $$ \frac{3}{2 x}-\frac{2}{3 x}=5 $$

A jar contains four blue marbles and two red marbles. Suppose you choose a marble at random, and do not replace it. Then you choose a second marble. Find the probability of each event. You select a red marble and then a blue marble.

Use the fact that \(\frac{b}{c}=a^{a} \div \frac{c}{d}\) to simplify each rational expression. State any restrictions on the variables. $$ \frac{\frac{3 a^{3} b^{3}}{a-b}}{\frac{4 a b}{b-a}} $$

Use this information for Exercises \(53-58\) . Bag 1 contains 5 red marbles, 1 blue marble, 3 yellow marbles, and 2 green marbles. Bag 2 contains 1 red pencil, 3 red pens, 2 blue pencils, and 5 blue pens. One marble is drawn from bag 1 . What is the probability that the marble is red or yellow?

Write each expression as a single logarithm. \(7 \log _{10} p+\log _{10} q\)
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