# Synthetic division of polynomials Calculator

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### Difficult Problems

1

Example

${\frac{x^3+5x+6}{x^2+x+42}}\geq {0}$
2

We can factor the polynomial $6+5x+x^3$ using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term $6$ and we found that $-1$ is a root of the polynomial

$6-1\cdot 5+{\left(-1\right)}^3=0$
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

$\left|\begin{array}{c}1 & 0 & 5 & 6 \\ & -1 & 1 & -6 \\ 1 & -1 & 6 & 0\end{array}\right|-1$
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 $x+1$

$\frac{\left(1+x\right)\left(x^{2}-x+6\right)}{42+x+x^2}\geq 0$
5

Multiplying polynomials $x$ and $6+-x$

$\frac{x^{2}-x+6+x^{2}x-x^2+6x}{42+x+x^2}\geq 0$
6

When multiplying exponents with same base you can add the exponents

$\frac{x^{2}-x+6+x^{3}-x^2+6x}{42+x+x^2}\geq 0$