# Synthetic division of polynomials Calculator

## Get detailed solutions to your math problems with our Synthetic division of polynomials step by step calculator. Sharpen your math skills and learn step by step with our math solver. Check out more online calculators here.

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### Difficult Problems

1

Solved example of Synthetic division of polynomials

$\left(-2x+2\right)\left(x^4+4x^3+6x^2+4x+1\right)+\left(x^2-2x-3\right)\left(4x^3+12x^2+12x+4\right)$
2

We can factor the polynomial $\left(x^4+4x^3+6x^2+4x+1\right)$ using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term $1$ and we found that $-1$ is a root of the polynomial

${\left(-1\right)}^4+4{\left(-1\right)}^3+6{\left(-1\right)}^2+4\cdot -1+1=0$
3

Let's divide the polynomial by $x+1$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $-1$. Add the result to the second coefficient and then multiply this by $-1$ and so on

$\left|\begin{array}{c}1 & 4 & 6 & 4 & 1 \\ & -1 & -3 & -3 & -1 \\ 1 & 3 & 3 & 1 & 0\end{array}\right|-1$
4

In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $x+1$

$\left(-2x+2\right)\left(x^{3}+3x^{2}+3x+1\right)\left(x+1\right)+\left(x^2-2x-3\right)\left(4x^3+12x^2+12x+4\right)$
5

We can factor the polynomial $\left(x^{3}+3x^{2}+3x+1\right)$ using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term $1$ and we found that $-1$ is a root of the polynomial

${\left(-1\right)}^{3}+3{\left(-1\right)}^{2}+3\cdot -1+1=0$
6

Let's divide the polynomial by $x+1$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $-1$. Add the result to the second coefficient and then multiply this by $-1$ and so on

$\left|\begin{array}{c}1 & 3 & 3 & 1 \\ & -1 & -2 & -1 \\ 1 & 2 & 1 & 0\end{array}\right|-1$
7

In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $x+1$

$\left(-2x+2\right)\left(x^{2}+2x+1\right)\left(x+1\right)\left(x+1\right)+\left(x^2-2x-3\right)\left(4x^3+12x^2+12x+4\right)$
8

When multiplying exponents with same base you can add the exponents

$\left(x+1\right)^2\left(-2x+2\right)\left(x^{2}+2x+1\right)+\left(x^2-2x-3\right)\left(4x^3+12x^2+12x+4\right)$
9

We can factor the polynomial $\left(4x^3+12x^2+12x+4\right)$ using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term $4$ and we found that $-1$ is a root of the polynomial

$4{\left(-1\right)}^3+12{\left(-1\right)}^2+12\cdot -1+4=0$
10

Let's divide the polynomial by $x+1$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $4$ and multiply by the factor $-1$. Add the result to the second coefficient and then multiply this by $-1$ and so on

$\left|\begin{array}{c}4 & 12 & 12 & 4 \\ & -4 & -8 & -4 \\ 4 & 8 & 4 & 0\end{array}\right|-1$
11

In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $x+1$

$\left(x+1\right)^2\left(-2x+2\right)\left(x^{2}+2x+1\right)+\left(x^2-2x-3\right)\left(4x^{2}+8x+4\right)\left(x+1\right)$
12

Multiplying polynomials $x^2$ and $4x^{2}x+4x^{2}$

$-2\left(x+1\right)^2x^{3}+2x^{2}\left(x+1\right)^2-4\left(x+1\right)^2x^2-2x\left(x+1\right)^2+4\left(x+1\right)^2x+2\left(x+1\right)^2+4x\cdot x^{4}+4x^{4}+x^{2}\left(-2x-3\right)\left(4x+4\right)+\left(x+1\right)\left(x^2-2x-3\right)\left(8x+4\right)$
13

Adding $-2x\left(x+1\right)^2$ and $4x\left(x+1\right)^2$

$-2\left(x+1\right)^2x^{3}+2x^{2}\left(x+1\right)^2-4\left(x+1\right)^2x^2+2\left(x+1\right)^2+4x\cdot x^{4}+4x^{4}+x^{2}\left(-2x-3\right)\left(4x+4\right)+\left(x+1\right)\left(x^2-2x-3\right)\left(8x+4\right)+2x\left(x+1\right)^2$
14

Multiplying polynomials $x$ and $-2x+-3$

$-2\left(x+1\right)^2x^{3}+2x^{2}\left(x+1\right)^2-4\left(x+1\right)^2x^2+2\left(x+1\right)^2+4x\cdot x^{4}+4x^{4}-8x^{4}-8x^{3}-12x^{3}-12x^{2}+8x^{4}+4x^{3}+8x^{3}+4x^2-16x^{3}-24x^2-8x^2-12x+\left(-2x-3\right)\left(8x+4\right)+2x\left(x+1\right)^2$
15

Adding $-32x^2$ and $-8x^2$

$-2\left(x+1\right)^2x^{3}+2\left(x+1\right)^2+4x\cdot x^{4}+8x^{4}-16x^{3}-12x+\left(-2x-3\right)\left(8x+4\right)+2x\left(x+1\right)^2-2x^{2}\left(x+1\right)^2-4x^{4}-8x^{3}-40x^2$
16

Expand $\left(x+1\right)^2$

$-2x^{3}\left(x^2+2x+1\right)+2\left(x^2+2x+1\right)+4x\cdot x^{4}+8x^{4}-16x^{3}-12x+\left(-2x-3\right)\left(8x+4\right)+2x\left(x^2+2x+1\right)+x^{2}\left(-2x^2-4x-2\right)-4x^{4}-8x^{3}-40x^2$
17

Solve the product $2\left(2x\cdot x+x\right)$

$x^{3}\left(-2x^2-4x-2\right)+2x^2+4x+2+4x\cdot x^{4}+8x^{4}-16x^{3}-12x+\left(-2x-3\right)\left(8x+4\right)+2x^{3}+4x^2+2x+x^{2}\left(-2x^2-4x-2\right)-4x^{4}-8x^{3}-40x^2$
18

Multiplying polynomials

$-2x^{3}x^2-4x^{3}x-2x^{3}+2x^2+4x+2+4x\cdot x^{4}+8x^{4}-16x^{3}-12x+\left(-2x-3\right)\left(8x+4\right)+2x^{3}+4x^2+2x-2x^{2}x^2-4x^{2}x-2x^{2}-4x^{4}-8x^{3}-40x^2$
19

When multiplying exponents with same base we can add the exponents

$-2x^{5}-4x^{3}x-2x^{3}+2x^2+4x+2+4x\cdot x^{4}+8x^{4}-16x^{3}-12x+\left(-2x-3\right)\left(8x+4\right)+2x^{3}+4x^2+2x-2x^{4}-4x^{2}x-2x^{2}-4x^{4}-8x^{3}-40x^2$
20

When multiplying exponents with same base you can add the exponents

$-2x^{5}-4x^{4}-2x^{3}+2x^2+4x+2+4x\cdot x^{4}+8x^{4}-16x^{3}-12x+\left(-2x-3\right)\left(8x+4\right)+2x^{3}+4x^2+2x-2x^{4}-4x^{3}-2x^{2}-4x^{4}-8x^{3}-40x^2$
21

Adding $-4x^{4}$ and $-4x^{4}$

$-2x^{5}-2x^{3}+2x^2+4x+2+4x\cdot x^{4}+8x^{4}-16x^{3}-12x+\left(-2x-3\right)\left(8x+4\right)+2x^{3}+4x^2+2x-2x^{4}-4x^{3}-2x^{2}-8x^{3}-40x^2-8x^{4}$
22

Adding $6x^2$ and $-42x^2$

$-2x^{5}+4x+2+4x\cdot x^{4}-12x+\left(-2x-3\right)\left(8x+4\right)+2x-2x^{4}-28x^{3}-36x^2$

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