Let's find the partial fraction decomposition of the rational expression:
[tex]\[
\frac{3}{x^2 + x - 2}
\][/tex]
### Step 1: Factor the Denominator
First, we need to factor the quadratic denominator [tex]\( x^2 + x - 2 \)[/tex]. We look for two numbers that multiply to [tex]\(-2\)[/tex] (the constant term) and add to [tex]\(1\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
The factorization is:
[tex]\[
x^2 + x - 2 = (x - 1)(x + 2)
\][/tex]
### Step 2: Set Up the Partial Fraction Decomposition
Given the factorization, we can express the original rational expression as the sum of two fractions with unknown coefficients:
[tex]\[
\frac{3}{x^2 + x - 2} = \frac{A}{x - 1} + \frac{B}{x + 2}
\][/tex]
### Step 3: Combine the Partial Fractions
We will combine the right side over a common denominator:
[tex]\[
\frac{A}{x - 1} + \frac{B}{x + 2} = \frac{A(x + 2) + B(x - 1)}{(x - 1)(x + 2)}
\][/tex]
### Step 4: Set Up the Equation
Since the denominators are the same, we can set the numerators equal to each other:
[tex]\[
3 = A(x + 2) + B(x - 1)
\][/tex]
### Step 5: Solve for the Coefficients [tex]\(A\)[/tex] and [tex]\(B\)[/tex]
Expand and combine like terms:
[tex]\[
3 = Ax + 2A + Bx - B
\][/tex]
[tex]\[
3 = (A + B)x + (2A - B)
\][/tex]
To solve for [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we equate the coefficients of [tex]\(x\)[/tex] and the constant terms on both sides:
1. [tex]\(A + B = 0\)[/tex]
2. [tex]\(2A - B = 3\)[/tex]
From the first equation:
[tex]\[
A = -B
\][/tex]
Substitute [tex]\(A = -B\)[/tex] into the second equation:
[tex]\[
2(-B) - B = 3
\][/tex]
[tex]\[
-2B - B = 3
\][/tex]
[tex]\[
-3B = 3
\][/tex]
[tex]\[
B = -1
\][/tex]
Then substituting [tex]\(B = -1\)[/tex] back into [tex]\(A = -B\)[/tex]:
[tex]\[
A = -(-1) = 1
\][/tex]
### Step 6: Write the Partial Fraction Decomposition
Now we have the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
A = 1 \quad \text{and} \quad B = -1
\][/tex]
Substitute [tex]\(A\)[/tex] and [tex]\(B\)[/tex] back into the partial fractions:
[tex]\[
\frac{3}{x^2 + x - 2} = \frac{1}{x - 1} - \frac{1}{x + 2}
\][/tex]
So, the partial fraction decomposition is:
[tex]\[
\frac{3}{x^2 + x - 2} = \frac{1}{x - 1} - \frac{1}{x + 2}
\][/tex]
### Step 7: Verify the Result
To verify, we can combine the partial fractions to check our answer:
[tex]\[
\frac{1}{x - 1} - \frac{1}{x + 2} = \frac{(x + 2) - (x - 1)}{(x - 1)(x + 2)} = \frac{x + 2 - x + 1}{(x - 1)(x + 2)} = \frac{3}{x^2 + x - 2}
\][/tex]
This confirms that our partial fraction decomposition is correct. The final answer is:
[tex]\[
\boxed{\frac{1}{x - 1} - \frac{1}{x + 2}}
\][/tex]
