# Polynomial long division Calculator

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### Difficult Problems

1

Solved example of Polynomial long division

$\int\frac{2x^3+6x^2-26x-54}{x^2+2x-15}dx$
2

Divide $2x^3+6x^2-26x-54$ by $x^2+2x-15$

$\begin{array}{l}\phantom{\phantom{;}x^{2}+2x\phantom{;}-15;}{\phantom{;}2x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+2x\phantom{;}-15\overline{\smash{)}\phantom{;}2x^{3}+6x^{2}-26x\phantom{;}-54\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}-15;}\underline{-2x^{3}-4x^{2}+30x\phantom{;}\phantom{-;x^n}}\\\phantom{-2x^{3}-4x^{2}+30x\phantom{;};}\phantom{;}2x^{2}+4x\phantom{;}-54\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+2x\phantom{;}-15-;x^n;}\underline{-2x^{2}-4x\phantom{;}+30\phantom{;}\phantom{;}}\\\phantom{;-2x^{2}-4x\phantom{;}+30\phantom{;}\phantom{;}-;x^n;}-24\phantom{;}\phantom{;}\\\end{array}$
3

Resulting polynomial

$\int\left(2x+2+\frac{-24}{x^2+2x-15}\right)dx$
4

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int2xdx+\int2dx+\int\frac{-24}{x^2+2x-15}dx$
5

The integral of a constant is equal to the constant times the integral's variable

$\int2xdx+2x+\int\frac{-24}{x^2+2x-15}dx$
6

Take the constant out of the integral

$2\int xdx+2x+\int\frac{-24}{x^2+2x-15}dx$
7

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$x^2+2x+\int\frac{-24}{x^2+2x-15}dx$
8

Factor the trinomial $x^2+2x-15$ finding two numbers that multiply to form $-15$ and added form $2$

$\begin{matrix}\left(-3\right)\left(5\right)=-15\\ \left(-3\right)+\left(5\right)=2\end{matrix}$
9

Thus

$x^2+2x+\int\frac{-24}{\left(x-3\right)\left(x+5\right)}dx$
10

Rewrite the fraction $\frac{-24}{\left(x-3\right)\left(x+5\right)}$ in $2$ simpler fractions using partial fraction decomposition

$\frac{-24}{\left(x-3\right)\left(x+5\right)}=\frac{A}{x-3}+\frac{B}{x+5}$
11

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x-3\right)\left(x+5\right)$

$-24=\left(x-3\right)\left(x+5\right)\left(\frac{A}{x-3}+\frac{B}{x+5}\right)$
12

Multiplying polynomials

$-24=\frac{A\left(x-3\right)\left(x+5\right)}{x-3}+\frac{B\left(x-3\right)\left(x+5\right)}{x+5}$
13

Simplifying

$-24=A\left(x+5\right)+B\left(x-3\right)$
14

Expand the polynomial

$-24=Ax+5A+Bx-3B$
15

Assigning values to $x$ we obtain the following system of equations

$\begin{matrix}-24=-6B+2A&\:\:\:\:\:\:\:(x=-3) \\ -24=8A&\:\:\:\:\:\:\:(x=3)\end{matrix}$
16

Proceed to solve the system of linear equations

$\begin{matrix}2A & - & 6B & =-24 \\ 8A & + & 0B & =-24\end{matrix}$
17

Rewrite as a coefficient matrix

$\left(\begin{matrix}2 & -6 & -24 \\ 8 & 0 & -24\end{matrix}\right)$
18

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & -3 \\ 0 & 1 & 3\end{matrix}\right)$
19

The decomposed integral equivalent is

$x^2+2x+\int\left(\frac{-3}{x-3}+\frac{3}{x+5}\right)dx$
20

The integral of a sum of two or more functions is equal to the sum of their integrals

$x^2+2x+\int\frac{-3}{x-3}dx+\int\frac{3}{x+5}dx$
21

Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|, where b=-3 and n=-3 x^2+2x-3\ln\left|x-3\right|+\int\frac{3}{x+5}dx 22 Apply the formula: \int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=5$ and $n=3$

$x^2+2x-3\ln\left|x-3\right|+3\ln\left|x+5\right|$
23

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

$x^2+2x-3\ln\left|x-3\right|+3\ln\left|x+5\right|+C_0$

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