Question

4X^2+13x-9
__________
X^3+2x^2-3x
rational expression as asum of partial fraction

Answer 1

The partial fraction decomposition
`(4x^2+13x-9)/(x(x+3)(x-1)) = (-3)/(x) + (3)/(x+3) + (0.5)/(x-1)`

First, factor the denominator to find the denominators of the two fractions we will split our rational expression into.

`x^3+2x^2-3x = x(x^2+2x-3) = x(x+3)(x-1)`

Since the denominator can be factored into three distinct linear factors, we can write out our rational expression as the sum of three fractions whose denominators are the individual factors we just found.

`(4x^2+13x-9)/(x(x+3)(x-1)) = (A)/(x) + (B)/(x+3) + (C)/(x-1)`

To find the numerators A, B, and C, we can use the method of undetermined coefficients. This method involves multiplying both sides of the equation by the common denominator, and then equating the coefficients of like terms on both sides.

`4x^2+13x-9 = A(x+3)(x-1) + B(x)(x-1) + C(x)(x+3)`

To solve for A, we set x = 0 and solve for A.

-9 = A(-3)(-1)

A = -3

To solve for B, we set x = -3 and solve for B.

28 = B(-3)(-4)

B = 3

To solve for C, we set x = 1 and solve for C.

2 = C(1)(4)

C = 0.5

Now that we have found A, B, and C, we can plug them back into our fractions to get the partial fraction decomposition.

`(4x^2+13x-9)/(x(x+3)(x-1)) = (-3)/(x) + (3)/(x+3) + (0.5)/(x-1)`