Question

Write out the form of the partial fraction decomposition of the function (See Example). Do not determine the numerical values of the coefficients. (If the partial fraction decomposition does not exist, enter DNE.)

(a) (x)/(x^2 + x − 6)

Answer 1

The partial fraction decomposition of the function (x)/(x^2 + x - 6) is A/(x + 3) + B/(x - 2), where A and B are coefficients that one would solve for separately.

Step-by-step explanation:

The function (x)/(x^2 + x - 6) can be decomposed into partial fractions when factoring the denominator. The denominator x^2 + x - 6 can be factored into (x + 3)(x - 2). The partial fraction decomposition of the function then has the form:

A/(x + 3) + B/(x - 2)

Here, A and B are the coefficients that will be determined by multiplying both sides of the equation by the common denominator to clear the fractions, then solving for the coefficients by equating the coefficients of like terms from both sides of the resulting equation.

Answer 2

I have a solution here for the same problem but a different given:

(2x-82)/(x^2+2x-48) dx

In which this solution will help you answer the problem by yourself:

(2x - 82) /(x² + 2x - 48)

first, factor the denominator completely:

(x² + 2x - 48) =

(x² + 8x - 6x - 48) =

x(x + 8) - 6(x + 8) =

(x - 6)(x + 8)

so the function becomes:

(2x - 82) /[(x - 6)(x + 8)]

let's decompose this into partial fractions:

(2x - 82) /[(x - 6)(x + 8)] = A/(x - 6) + B/(x + 8)

(letting [(x - 6)(x + 8)] be the common denominator at the right side too)

(2x - 82) /[(x - 6)(x + 8)] = [A(x + 8) + B(x - 6)] /[(x - 6)(x + 8)]

(equating numerators)

2x - 82 = Ax + 8A + Bx - 6B

2x - 82 = (A + B)x + (8A - 6B)

yielding the system:

A + B = 2
8A - 6B = - 82

A = 2 - B
4A - 3B = - 41

A = 2 - B
4(2 - B) - 3B = - 41

A = 2 - B
8 - 4B - 3B = - 41

A = 2 - B
- 7B = - 41 - 8

A = 2 - B
7B = 49

A = 2 - 7 = - 5
B = 49/7 = 7

hence:

(2x - 82) /[(x - 6)(x + 8)] = A/(x - 6) + B/(x + 8) = - 5/(x - 6) + 7/(x + 8)

thus the answer is:


(2x - 82) /(x² + 2x - 48) = [- 5 /(x - 6)] + [7 /(x + 8)]