Question

What is the correct form of the partial fraction decomposition for the expression a. StartFraction 13 x minus 7 Over (x + 2) (4 x minus 3) EndFraction?

b. StartFraction A Over x + 2 EndFraction + StartFraction B Over 4 x minus 3 EndFraction
c. StartFraction A Over x + 2 EndFraction + StartFraction B x + C Over 4 x minus 3 EndFraction
d. StartFraction A Over (x + 2) (4 x minus 3) EndFraction + StartFraction B Over (x + 2) (4 x minus 3) EndFraction
e. StartFraction A Over (x + 2) (4 x minus 3) EndFraction + StartFraction B x + C Over (x + 2) (4 x minus 3) EndFraction

Answer 1

The correct form of the partial fraction decomposition for the given expression 13x - 7 / x + 2 / 4x - 3 is:

c. A / x + 2 + Bx + C / 4x - 3

Step-by-step explanation:

To perform partial fraction decomposition, you express a rational function as the sum of simpler fractions. In this case, the given expression is 13x - 7 / (x + 2)(4x - 3).

The decomposition is written as:

A / x + 2 + Bx + C / 4x - 3

Here's the rationale:

1. A is the constant in the numerator for the factor x + 2.

2. Bx + C is a linear expression in the numerator for the factor 4x - 3.

By finding the values of A, B, and C, you can rewrite the original expression in this partial fraction form.

This specific choice (option c) is the correct one, as it correctly decomposes the given expression into the sum of two fractions with the appropriate forms for each factor in the denominator.

The correct option is c.