Question

Divide 6x^6+5x^5+2x^4-9x^3+7x^2-10x+2 by 3x + 1 by using long division. Show all work and steps work. Then explain if 3x + 1 is a factor of the dividend.

Answer 1

A) First, write all the terms of the polynomial separated; then follow these steps:

1) take the term of the highest degree of the dividend and divide it by the term of the highest degree of the divisor (6x⁶ ÷ 3x = 2x⁵) and write the result on the proper part of the division;

2) multiply the monomial obtained times the divisor (2x⁵ × 3x+1 = 6x⁶ + 2x⁵), write the result under the polynomial, under the colum of the same degree;

3) subtract the result obtained from the polynomial (all the terms of the polynomial!!);

4) repeat step 1-3 with the new polynomial obtained.

Complete work is shown on the picture attached.

The result is: 6x⁶ + 5x⁵ + 2x⁴ - 9x³ + 7x² -10x + 2 = (3x + 1) · (2x⁵ + x⁴ + 1/3 x³ - 28/9 x² + 91/27 x - 361/81) + 523/81

B) (3x + 1) is not a factor of the dividend because the division has a remainder not equal to zero.