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

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Answer 1 · 2016-12-11T01:42:23

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

Divide the numerator by the denominator using long division.

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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!

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