Chapter7: Rational Expressions And Functions
Section7.3: Simplify Complex Rational Expressions
Problem 195E: In this section, you learned to simplify the complex fraction 3x+2xx24 two ways: rewriting it as a...
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![### Polynomial Fraction Addition
In this example, we demonstrate how to add two polynomial fractions together.
The given expression is:
\[
\frac{3x^2 - 5x - 2}{2x + 14} + \frac{x^2 + x - 2}{3x + 21}
\]
**Steps to Solve:**
1. Analyze the denominators \(2x + 14\) and \(3x + 21\).
2. Factorize the denominators wherever possible.
3. Find a common denominator.
4. Re-write each fraction with the common denominator.
5. Add the numerators and simplify the resulting polynomial if possible.
Let's factorize the denominators:
\[
2x + 14 = 2(x + 7)
\]
\[
3x + 21 = 3(x + 7)
\]
The common denominator will be \(6(x + 7)\).
Next, rewrite each fraction with the common denominator:
\[
\frac{3x^2 - 5x - 2}{2(x + 7)} = \frac{3(3x^2 - 5x - 2)}{6(x + 7)}
\]
\[
\frac{x^2 + x - 2}{3(x + 7)} = \frac{2(x^2 + x - 2)}{6(x + 7)}
\]
Combine the fractions:
\[
\frac{3(3x^2 - 5x - 2) + 2(x^2 + x - 2)}{6(x + 7)}
\]
Now, simplify the numerator:
\[
3(3x^2 - 5x - 2) = 9x^2 - 15x - 6
\]
\[
2(x^2 + x - 2) = 2x^2 + 2x - 4
\]
\[
9x^2 - 15x - 6 + 2x^2 + 2x - 4 = 11x^2 - 13x - 10
\]
The combined fraction is:
\[
\frac{11x^2 - 13x - 10}{6(x + 7)}
\]
Hence, the result of adding the two polynomial fractions is:
\[
\frac{11x^2 -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe78f7d0f-626d-4ef1-bc58-30bf6b75a3d5%2Fabf44768-8be9-4489-8fe4-e9ef0a43c055%2Fl86pl3ih_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomial Fraction Addition
In this example, we demonstrate how to add two polynomial fractions together.
The given expression is:
\[
\frac{3x^2 - 5x - 2}{2x + 14} + \frac{x^2 + x - 2}{3x + 21}
\]
**Steps to Solve:**
1. Analyze the denominators \(2x + 14\) and \(3x + 21\).
2. Factorize the denominators wherever possible.
3. Find a common denominator.
4. Re-write each fraction with the common denominator.
5. Add the numerators and simplify the resulting polynomial if possible.
Let's factorize the denominators:
\[
2x + 14 = 2(x + 7)
\]
\[
3x + 21 = 3(x + 7)
\]
The common denominator will be \(6(x + 7)\).
Next, rewrite each fraction with the common denominator:
\[
\frac{3x^2 - 5x - 2}{2(x + 7)} = \frac{3(3x^2 - 5x - 2)}{6(x + 7)}
\]
\[
\frac{x^2 + x - 2}{3(x + 7)} = \frac{2(x^2 + x - 2)}{6(x + 7)}
\]
Combine the fractions:
\[
\frac{3(3x^2 - 5x - 2) + 2(x^2 + x - 2)}{6(x + 7)}
\]
Now, simplify the numerator:
\[
3(3x^2 - 5x - 2) = 9x^2 - 15x - 6
\]
\[
2(x^2 + x - 2) = 2x^2 + 2x - 4
\]
\[
9x^2 - 15x - 6 + 2x^2 + 2x - 4 = 11x^2 - 13x - 10
\]
The combined fraction is:
\[
\frac{11x^2 - 13x - 10}{6(x + 7)}
\]
Hence, the result of adding the two polynomial fractions is:
\[
\frac{11x^2 -
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