# Step-by-step Solution

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## Step-by-step explanation

Problem to solve:

$\frac{3x^5+7x^4-12x^3+40x^2+24x-32}{2x^3-2x^2+4x+8}$

Learn how to solve polynomial long division problems step by step online.

$2\cdot {\left(-1\right)}^3-2\cdot {\left(-1\right)}^2+4\cdot -1+8=0$

Learn how to solve polynomial long division problems step by step online. Simplify the expression (3x^5+7x^4-12x^3+40x^2+24x-32)/(2x^3-2x^2+4x+8). We can factor the polynomial 2x^3-2x^2+4x+8 using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term 8 and we found that -1 is a root of the polynomial. 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 2 and multiply by the factor -1. Add the result to the second coefficient and then multiply this by -1 and so on. 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. We can factor the polynomial 3x^5+7x^4-12x^3+40x^2+24x-32 using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term -32 and we found that -1 is a root of the polynomial.

$\frac{3}{2}\left(-\frac{49}{9}+\left(x+\frac{5}{3}\right)^2\right)$

### Problem Analysis

$\frac{3x^5+7x^4-12x^3+40x^2+24x-32}{2x^3-2x^2+4x+8}$

### Main topic:

Polynomial long division

~ 0.09 seconds