Question

Review the work showing the first few steps in writing a partial fraction decomposition.

StartFraction 4 x + 40 Over (x + 2) (x + 6) EndFraction = StartFraction A Over x + 2 EndFraction + StartFraction 16 Over x + 6 EndFraction
4x + 40 = A(x + 6) + B(x + 2)
4x + 40 = Ax + 6A + Bx + 2B

What is the partial fraction decomposition in terms of x?

Answer 1

To write a partial fraction decomposition, we need to decompose the given expression into two separate fractions with different denominators. By multiplying both sides of the equation by the common denominator and comparing the coefficients of x, we can determine the values of A and B. The partial fraction decomposition of the expression is then obtained by expressing it as the sum of the fractions with their respective coefficients.

Step-by-step explanation:

To write a partial fraction decomposition for the expression 4x + 40 / ((x + 2)(x + 6)), we need to decompose it into two separate fractions with different denominators. We can rewrite the expression as:

4x + 40 = A / (x + 2) + B / (x + 6)

We can then multiply both sides of the equation by the common denominator, which in this case is (x + 2)(x + 6). This eliminates the denominators and leaves us with:

(4x + 40)(x + 2)(x + 6) = A(x + 6) + B(x + 2)

Expanding and simplifying, we get:

4x^2 + 76x + 240 = Ax + 6A + Bx + 2B

By comparing the coefficients of x on both sides of the equation, we can determine that A = 16 and B = 4.

Therefore, the partial fraction decomposition of the given expression is:

4x + 40 / ((x + 2)(x + 6)) = 16 / (x + 2) + 4 / (x + 6)