Question

Perform partial fraction decomposition for x² / (x² + 12).

A. 1 - 12 / (x² + 12)
B. 1 + 12 / (x² + 12)
C. 1 / x² - 12 / 12
D. 1 / (x² + 12) - 12 / 12

Answer 1

The partial fraction decomposition of x² / (x² + 12) is 1 - 12 / (x² + 12), achieved by subtracting 12 from the numerator and denominator, simplifying the fraction without the need for complex algebraic manipulation. Option a is the correct answer.

Step-by-step explanation:

To perform partial fraction decomposition for the expression x² / (x² + 12), we can start by noting that the numerator is simply the denominator less 12. Therefore, the expression can be re-written as:

(x² + 12 - 12) / (x² + 12) = 1 - 12 / (x² + 12)

Thus, the correct form of the partial fraction decomposition of x² / (x² + 12) is:

1 - 12 / (x² + 12)

No complex algebraic rearrangement or quadratic formula is necessary for this decomposition. It is essential to understand the structure of fractions and how numerators and denominators relate to simplify the given expression correctly.