# How do you divide 3x^4+5x^3-x^2+x-2  by x-2?

See in the explanation

#### Explanation:

We have that

$3 {x}^{4} + 5 {x}^{3} - {x}^{2} + x - 2 = \left(3 {x}^{3} + 11 {x}^{2} + 21 x + 43\right) \cdot \left(x - 2\right) + 84$

Sep 11, 2015

See the explanation section.

#### Explanation:

$\frac{3 {x}^{4} + 5 {x}^{3} - {x}^{2} + x - 2}{x - 2}$

There are various ways of writing the details Here's one way.

$\text{ " " }$$- - - - - - - -$
x-2 ) $3 {x}^{4}$ $+ 5 {x}^{3}$ $- {x}^{2}$ $+ x$ $- 2$

What do we need to multiply the first term on the divisor ($x$) by to get the first term of the dividend ($3 {x}^{4}$)? Clearly, we need to multiply by $3 {x}^{3}$

$\text{ " " " " " } 3 {x}^{3}$
$\text{ " " }$$- - - - - - - -$
x-2 ) $3 {x}^{4}$ $+ 5 {x}^{3}$ $- {x}^{2}$ $+ x$ $- 2$

Now multiply $3 {x}^{3}$ times the divisor, $x - 2$, to get $3 {x}^{4} - 6 {x}^{3}$ and write that under the dividend.

$\text{ " " " " " } 3 {x}^{3}$
$\text{ " " }$$- - - - - - - -$
x-2 ) $3 {x}^{4}$ $+ 5 {x}^{3}$ $- {x}^{2}$ $+ x$ $- 2$
$\text{ "" "" }$ $3 {x}^{4}$ $- 6 {x}^{3}$
$\text{ " " }$$- - - - -$

Now we need to subtract $3 {x}^{4} - 6 {x}^{3}$ from the dividend. (You may find it simpler to change the signs and add.)

$\text{ " " " " " "" } 3 {x}^{3}$
$\text{ " " }$$- - - - - - - -$
x-2 )" " $3 {x}^{4}$ $+ 5 {x}^{3}$ $- {x}^{2}$ $+ x$ $- 2$
$\text{ " " }$ $\textcolor{red}{-} 3 {x}^{4} \textcolor{red}{+} 6 {x}^{3}$
$\text{ "" "" }$$- - - - -$
$\text{ "" "" "" "" }$ $11 {x}^{3}$$- {x}^{2}$ $+ x$ $- 2$

Now, what do we need to multiply $x$ (the first term of the divisor) by to get $11 {x}^{3}$ (the first term of the last line)? We need to multiply by $11 {x}^{2}$
So write $11 {x}^{2}$ on the top line, then multiply $11 {x}^{2}$ times the divisor $x - 2$, to get $11 {x}^{3} - 22 {x}^{2}$ and write it underneath.

$\text{ " " " " " "" } 3 {x}^{3}$ $+ 11 {x}^{2}$
$\text{ " " }$$- - - - - - - -$
x-2 )" " $3 {x}^{4}$ $+ 5 {x}^{3}$ $- {x}^{2}$ $+ x$ $- 2$
$\text{ " " }$ $\textcolor{red}{-} 3 {x}^{4} \textcolor{red}{+} 6 {x}^{3}$
$\text{ "" "" }$$- - - - -$
$\text{ "" "" "" "" }$ $11 {x}^{3}$$- {x}^{2}$ $+ x$ $- 2$
$\text{ "" "" "" "" }$ $11 {x}^{3}$$- 22 {x}^{2}$
$\text{ " " "" "" }$$- - - - - -$

Now subtract (change the signs and add), to get:

$\text{ " " " " " "" } 3 {x}^{3}$ $+ 11 {x}^{2}$
$\text{ " " }$$- - - - - - - -$
x-2 )" " $3 {x}^{4}$ $+ 5 {x}^{3}$ $- {x}^{2}$$\text{ }$ $+ x$ $- 2$
$\text{ " " }$ $\textcolor{red}{-} 3 {x}^{4} \textcolor{red}{+} 6 {x}^{3}$
$\text{ "" "" }$$- - - - -$
$\text{ "" "" "" "" }$ $11 {x}^{3}$$- {x}^{2}$$\text{ }$ $+ x$ $- 2$
$\text{ "" "" "" }$ $\textcolor{red}{-} 11 {x}^{3}$$\textcolor{red}{+} 22 {x}^{2}$
$\text{ " " "" "" }$$- - - - - -$
$\text{ "" "" "" "" "" "" "" }$ $21 {x}^{2}$  +x -2

Repeat to get $21 x$, so we put the $9$ on top multiply, subtract (change signs and add) to get:

$\text{ " " " " " "" } 3 {x}^{3}$ $+ 11 {x}^{2}$ $+ 21 x$
$\text{ " " }$$- - - - - - - -$
x-2 )" " $3 {x}^{4}$ $+ 5 {x}^{3}$ $- {x}^{2}$$\text{ }$ $+ x$ $- 2$
$\text{ " " }$ $\textcolor{red}{-} 3 {x}^{4} \textcolor{red}{+} 6 {x}^{3}$
$\text{ "" "" }$$- - - - -$
$\text{ "" "" "" "" }$ $11 {x}^{3}$$- {x}^{2}$$\text{ }$ $+ x$ $- 2$
$\text{ "" "" "" }$ $\textcolor{red}{-} 11 {x}^{3}$$\textcolor{red}{+} 22 {x}^{2}$
$\text{ " " "" "" }$$- - - - - -$
$\text{ "" "" "" "" "" "" "" }$ $21 {x}^{2}$  +x" " $- 2$
$\text{ "" "" "" "" "" "" }$ $\textcolor{red}{-} 21 {x}^{2}$ $\textcolor{red}{+} 42 x$
$\text{ " " "" "" "" "" }$$- - - - - - - -$
$\text{ "" "" "" "" "" "" "" "" "" "" }$$43 x$ $- 2$
We'll be done when the last line is $0$ or has degree less than the degree of the divisor. Which has not happened yet, but we're close.

$\text{ " " " " " "" } 3 {x}^{3}$ $+ 11 {x}^{2}$ $+ 21 x$ $+ 43$
$\text{ " " }$$- - - - - - - -$
x-2 )" " $3 {x}^{4}$ $+ 5 {x}^{3}$ $- {x}^{2}$$\text{ }$ $+ x$ $- 2$
$\text{ " " }$ $\textcolor{red}{-} 3 {x}^{4} \textcolor{red}{+} 6 {x}^{3}$
$\text{ "" "" }$$- - - - -$
$\text{ "" "" "" "" }$ $11 {x}^{3}$$- {x}^{2}$$\text{ }$ $+ x$ $- 2$
$\text{ "" "" "" }$ $\textcolor{red}{-} 11 {x}^{3}$$\textcolor{red}{+} 22 {x}^{2}$
$\text{ " " "" "" }$$- - - - - -$
$\text{ "" "" "" "" "" "" "" }$ $21 {x}^{2}$  +x" " $- 2$
$\text{ "" "" "" "" "" "" }$ $\textcolor{red}{-} 21 {x}^{2}$ $\textcolor{red}{+} 42 x$
$\text{ " " "" "" "" "" }$$- - - - - - - -$
$\text{ "" "" "" "" "" "" "" "" "" "" }$$43 x$ $- 2$
$\text{ "" "" "" "" "" "" "" "" "" }$$\textcolor{red}{-} 43 x$ $\textcolor{red}{+} 86$
$\text{ " " "" "" "" "" }$$- - - - - - - -$
$\text{ "" "" "" "" "" "" "" "" "" "" "" "" "" }$ $84$

Now the last line has degree less than $1$, so we are finished.

The quotient is: $3 {x}^{3} + 11 {x}^{2} + 21 x + 43$ and the remainder is $84$

We can write:

$\frac{3 {x}^{4} + 5 {x}^{3} - {x}^{2} + x - 2}{x - 2} = 3 {x}^{3} + 11 {x}^{2} + 21 x + 43 + \frac{84}{x - 2}$

IMPORTANT to understanding what we have done:
If we get a common denominator on the right and simplify we will get exactly the left side.