Simplifying Rational Expressions and Partial Fraction Decomposition
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3. Find the domain and perform each of the following indicated operations for:
a. $\frac{x-1}{x+2} + \frac{x+3}{x^2+6x+5}$
b. $\frac{x-1}{x+2} - \frac{x+3}{x^2+6x+5}$
c. $\frac{(x-1)}{(x+2)} \times \frac{(x+3)}{(x^2+6x+5)}$
d. $\frac{x-1}{x+2} \div \frac{x+3}{x^2+6x+5}$
4. Write each of the following rational expressions as sum of partial fractions:
a. $\frac{5x+6}{x^2-4}$
b. $\frac{x+6}{x^2-4x+4}$
c. $\frac{6x+5}{x^4+4x^2}$
d. $\frac{x^2+x+2}{x^2+6x+8}$
e. $\frac{x^3+2x^2+2}{x^2-x-6}$
5. Solve each of the following rational equations.
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Today we are going to perform the operation indicated for this rational expression and find its domain. We have two rational expressions multiplied together.
Multiplying Rational Expressions
To simplify this, we should first factor all the polynomials in the numerators and denominators.
Step 1: Factor everything
The numerator x minus one, the denominator x plus two, and the second numerator x plus three are already in their simplest linear form. Let's focus on the quadratic expression x squared plus six x plus five.
We need two numbers that multiply to five and add up to six. Those numbers are five and one.
Now, let's rewrite the original expression using the factored form of the denominator.
Before we multiply, we must determine the domain. The domain consists of all real values of x except those that make any denominator zero.
Finding the Domain
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