Question

Review the work showing the first few steps in writing a partial fraction decomposition. start fraction x cubed 8 x minus 3 over (x squared 5) squared end fraction = start fraction a x b over x squared 5 end fraction start fraction c x d over (x squared 5) squared end fraction x3 8x – 3 = (ax b)(x2 5) cx d x3 8x – 3 = ax3 5ax bx2 5b cx d what are the values of a, b, c, and d?

a. a = 0; b = 1; c = –3; d = 3
b. a = 0; b = 1; c = 3; d = –3
c. a = 1; b = 0; c = –3; d = 3
d. a = 1; b = 0; c = 3; d = –3

Answer 1

After equating the coefficients on both sides of the given equation for partial fraction decomposition, we find that a and b must be zero because there are no x³ or x² terms in the original numerator. Since there is no x term, c must also be zero, leaving d = -3. None of the provided options are fully correct based on this decomposition.

Step-by-step explanation:

To solve for the constants a, b, c, and d in the partial fraction decomposition of x³ + 8x - 3 over (x² + 5)², we must equate coefficients of like powers of x on both sides of the equation:

x³ + 8x - 3 = (ax + b)(x² + 5) + ​​(cx + d) over (x² + 5)²

To find the correct values of a, b, c, and d, let's expand the right side of the equation and compare the coefficients with those on the left side:

• a must be 0, as there's no x³ term on the left-hand side.
• b is the coefficient of x², which also must be 0 since there's no such term on the left-hand side.
• The constant terms must match, giving us the equation c x + d = -3. Without an x term on the left, c must also be 0.
• Thus, d must be -3.

Following this logic, the correct answer is a = 0, b = 0, c = 0, and d = -3. However, there appears to be a mistake in the provided options, as none of them is fully correct based on our decomposition.