Sure, let's determine the partial fraction decomposition of the given rational expression step by step.
### Expression:
[tex]\[ \frac{17x - 53}{x^2 - 2x - 15} \][/tex]
#### Step 1: Factor the denominator
First, we need to factor the denominator [tex]\(x^2 - 2x - 15\)[/tex].
The quadratic [tex]\(x^2 - 2x - 15\)[/tex] can be factored into:
[tex]\[ x^2 - 2x - 15 = (x - 5)(x + 3) \][/tex]
So, the expression becomes:
[tex]\[ \frac{17x - 53}{(x - 5)(x + 3)} \][/tex]
#### Step 2: Set up the partial fraction decomposition
The next step is to express the fraction as the sum of simpler fractions:
[tex]\[ \frac{17x - 53}{(x - 5)(x + 3)} = \frac{A}{x - 5} + \frac{B}{x + 3} \][/tex]
where [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are constants to be determined.
#### Step 3: Combine the right-hand side into a single fraction
We need a common denominator on the right-hand side:
[tex]\[ \frac{A}{x - 5} + \frac{B}{x + 3} = \frac{A(x + 3) + B(x - 5)}{(x - 5)(x + 3)} \][/tex]
Equate the numerators:
[tex]\[ 17x - 53 = A(x + 3) + B(x - 5) \][/tex]
#### Step 4: Expand and collect like terms
Expand the right-hand side:
[tex]\[ 17x - 53 = Ax + 3A + Bx - 5B \][/tex]
Combine like terms:
[tex]\[ 17x - 53 = (A + B)x + (3A - 5B) \][/tex]
#### Step 5: Equate coefficients and solve for [tex]\(A\)[/tex] and [tex]\(B\)[/tex]
We now have a system of linear equations by equating the coefficients of [tex]\(x\)[/tex] and the constant terms on both sides:
For the coefficients of [tex]\(x\)[/tex]:
[tex]\[ A + B = 17 \][/tex]
For the constant terms:
[tex]\[ 3A - 5B = -53 \][/tex]
Solving this system, we find:
1. From [tex]\(A + B = 17\)[/tex], we can express [tex]\(A\)[/tex] in terms of [tex]\(B\)[/tex]:
[tex]\[ A = 17 - B \][/tex]
2. Substitute [tex]\(A = 17 - B\)[/tex] into the second equation:
[tex]\[ 3(17 - B) - 5B = -53 \][/tex]
[tex]\[ 51 - 3B - 5B = -53 \][/tex]
[tex]\[ 51 - 8B = -53 \][/tex]
[tex]\[ -8B = -104 \][/tex]
[tex]\[ B = 13 \][/tex]
3. Substitute [tex]\(B = 13\)[/tex] back into [tex]\(A = 17 - B\)[/tex]:
[tex]\[ A = 17 - 13 \][/tex]
[tex]\[ A = 4 \][/tex]
#### Step 6: Write the partial fractions
Thus, the partial fraction decomposition is:
[tex]\[ \frac{17x - 53}{(x - 5)(x + 3)} = \frac{4}{x - 5} + \frac{13}{x + 3} \][/tex]
### Final Answer:
[tex]\[ \frac{17x - 53}{x^2 - 2x - 15} = \frac{4}{x - 5} + \frac{13}{x + 3} \][/tex]
