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Chapter 8: Problem 4

Perform the indicated operations. $$\frac{x+3}{x^{2}+x+1}+\frac{1}{x-1}$$

Short Answer

Expert verified
Combine the numerators and write the single fraction: \ \frac{2x^2 + 3x - 2}{(x^2 + x + 1)(x - 1)} \.

Step by step solution

01

- Find a Common Denominator

To add the fractions, find a common denominator for both. The denominators here are \(x^2 + x + 1\) and \(x - 1\). The common denominator would be the product of these two: \((x^2 + x + 1)(x - 1)\).
02

- Rewrite Each Fraction with the Common Denominator

Rewrite each fraction so they have the common denominator \((x^2 + x + 1)(x - 1)\). This gives us: \[ \frac{(x+3)(x-1)}{(x^2 + x + 1)(x - 1)} + \frac{x^2 + x + 1}{(x^2 + x + 1)(x - 1)} \].
03

- Expand and Simplify the Numerators

Expand the numerators: \[ (x + 3)(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3 \] and \[ (x^2 + x + 1) \]. Add the simplified numerators: \[ (x^2 + 2x - 3) + (x^2 + x + 1) = 2x^2 + 3x - 2 \].
04

- Combine the Fractions

Now combine the fractions into a single fraction: \[ \frac{2x^2 + 3x - 2}{(x^2 + x + 1)(x - 1)} \]

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Key Concepts

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

Common Denominator
When working with fractions, a common denominator is crucial for performing addition or subtraction.

In this problem, we need to find a common denominator for the fractions \(\frac{x+3}{x^{2}+x+1} + \frac{1}{x-1}\). The denominators are \(x^2 + x + 1\) and \(x - 1\).

The common denominator will be the product of these two polynomial expressions: \( (x^2 + x + 1)(x - 1) \).

Finding this common denominator is the first step toward making the fractions comparable.
Fraction Addition
To add two fractions, they must have the same denominator.

With the common denominator \( (x^2 + x + 1)(x - 1) \), we rewrite each fraction so that they both share this denominator.

\[ \frac{(x+3)(x-1)}{(x^2 + x + 1)(x - 1)} + \frac{x^2 + x + 1}{(x^2 + x + 1)(x - 1)} \]

This makes it possible to combine the fractions by adding their numerators while keeping the common denominator.
Simplification of Rational Expressions
Simplifying the numerators involves expanding and combining like terms.

The process starts with expanding \( (x + 3)(x - 1) \):

\[ (x + 3)(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3 \]

Next, add this to the numerator of the second fraction, \(x^2 + x + 1\):

\[ (x^2 + 2x - 3) + (x^2 + x + 1) = 2x^2 + 3x - 2 \]

The simplified numerator is \(2x^2 + 3x - 2\).
Polynomial Expansion
Polynomial expansion involves distributing and combining like terms.

When you expand \( (x + 3)(x - 1) \), the process looks like this:

- Multiply each term in the first polynomial by each term in the second polynomial.
- Combine the resulting terms.

For example,

\[ (x + 3)(x - 1) \]

Multiply \(x \) by \(x \) to get \(x^2 \).
Multiply \(x \) by \(-1 \) to get \(-x \).
Multiply \(3 \) by \(x \) to get \(3x \).
Multiply \(3 \) by \(-1 \) to get \(-3 \).

Combining these, you get \( x^2 + 2x - 3 \). This step is essential for simplifying and combining rational functions.

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Most popular questions from this chapter

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