Question

How do you complete the following division: `(x^4 + x^3 - 5x^2 + 26x - 21)/(x^2 + 3x - 4)`?

Answer 1

`x^2 + 2x + 5 + 13/(5(x + 4)) + 2/(5(x - 1))`

Explanation:

Divide the numerator by the denominator using long division.

So, `(x^4 + x^3 - 5x^2 + 26x - 21)/(x^2 + 3x - 4) = x^2 + 2x + 5 + (3x - 1)/(x^2 + 3x - 4)`.

We can now start the actual partial fraction decomposition process.

`x^2 + 3x - 4` can be factored as `(x +4)(x -1)`.

`A/(x + 4) + B/(x- 1) = (3x -1)/((x+ 4)(x - 1))`

`A(x - 1) + B(x +4) = 3x - 1`

`Ax - A + Bx + 4B = 3x - 1`

`(A + B)x + (4B - A) = 3x - 1`

We now write a system of equations:

`{(A + B = 3), (4B - A= -1):}`

Solve:

`B = 3 - A -> 4(3 - A) - A = -1`

`12 - 4A - A = -1`

`-5A = -13`

`A = 13/5`

`13/5 + B = 3`

`B = 2/5`

Therefore, the partial fraction decomposition of `(x^4 + x^3 - 5x^2 + 26x - 21)/(x^2 + 3x - 4)` is `x^2 + 2x + 5 + 13/(5(x + 4)) + 2/(5(x - 1))`.

Hopefully this helps!