How do you divide $\frac{- 3 x^{3} - 3 x^{2} - 4 x + 1}{3 x - 4}$ $(-3x^3-3x^2-4x+1)/(3x-4)$ ?

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How do you divide $\frac{- 3 x^{3} - 3 x^{2} - 4 x + 1}{3 x - 4}$ $(-3x^3-3x^2-4x+1)/(3x-4)$ ?

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Answer 1 · 2017-10-15T15:44:52

I prefer long division.

Explanation:

$\frac{- 3 x^{3} - 3 x^{2} - 4 x + 1}{3 x - 4}$ $(-3x^3-3x^2-4x+1)/(3x-4)$

In long division form:

$\frac{3 x - 4}{3 x - 4} \frac{(- 3 x^{3} - 3 x^{2} - 4 x + 1)}{) - 3 x^{3} - 3 x^{2} - 4 x + 1}$ $color(white)( (3x-4)/color(black)(3x-4)) color(white)( (-3x^3-3x^2-4x+1))/(") "-3x^3-3x^2-4x+1)$

Write $- x^{2}$ $-x^2$ in the quotient:

$\frac{3 x - 4}{3 x - 4} \frac{.... - x^{2} - 3 x^{2} - 4 x + 1}{) - 3 x^{3} - 3 x^{2} - 4 x + 1}$ $color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2color(white)(-3x^2-4x+1))/(") "-3x^3-3x^2-4x+1)$

Multiply $- x^{2} (3 x - 4) = - 3 x^{2} + 4 x^{2}$ $-x^2(3x-4) = -3x^2+ 4x^2$ and subtract from the dividend:

$\frac{3 x - 4}{3 x - 4} \frac{.... - x^{2} - 3 x^{2} - 4 x + 1}{) - 3 x^{3} - 3 x^{2} - 4 x + 1}$ $color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2color(white)(-3x^2-4x+1))/(") "-3x^3-3x^2-4x+1)$
$color(white)(".....................")ul(3x^2- 4x^2)$
$color(white)(".................... .")-7x^2-4x$

Write $-7/3x$ in the quotient:

$color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2-7/3xcolor(white)(-4x+1))/(") "-3x^3-3x^2-4x+1)$
$color(white)(".....................")ul(3x^2- 4x^2)$
$color(white)(".................... .")-7x^2-4x$

Multiply $-7/3x(3x-4) = -7x^2+ 28/3x$ and subtract from the dividend:

$color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2-7/3xcolor(white)(-4x+1))/(") "-3x^3-3x^2-4x+1)$
$color(white)(".....................")ul(3x^2- 4x^2)$
$color(white)(".................... .")-7x^2-4x$
$color(white)("....................... .")ul(7x^2-28/3x)$
$color(white)("............................... .")-40/3x+ 1$

Write $-40/9$ in the quotient:

$color(white)( (3x-4)/color(black)(3x-4)) (color(white)("...")-x^2-7/3x-40/9color(white)(1))/(") "-3x^3-3x^2-4x+1)$
$color(white)(".....................")ul(3x^2- 4x^2)$
$color(white)(".................... .")-7x^2-4x$
$color(white)("....................... .")ul(7x^2-28/3x)$
$color(white)("............................... .")-40/3x+ 1$

Multiply $-40/9(3x-4) = -40/9x+ 160/9$ and subtract from the dividend:

$color(white)( (3x-4)/color(black)(3x-4)) (color(white)("...")-x^2-7/3x-40/9color(white)(1))/(") "-3x^3-3x^2-4x+1)$
$color(white)(".....................")ul(3x^2- 4x^2)$
$color(white)(".................... .")-7x^2-4x$
$color(white)("....................... .")ul(7x^2-28/3x)$
$color(white)("............................... .")-40/3x+ 1$
$color(white)("............................... .")ul(40/9x - 160/9)$
$color(white)("........................................... .")-151/9$

The quotient is $-x^2-7/3x-40/9$ with a remainder of $-151/9$

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