# Polynomial division

• Dec 13th 2011, 02:48 AM
luke11121
Polynomial division
hey guys

Im fairly sure ive never been taught this and i have a few assignment questions to complete and im struggling to get my head around it.

The first one is 4x^2 - 3x - 2 divided by 1 - X

Can anyone give me some tips? hopefully from this first i'll manage the rest

Thanks
• Dec 13th 2011, 03:03 AM
emakarov
Re: Polynomial division
This Wikipedia page has a detailed example of long polynomial division. If you just need the remainder, you can use little Bézout's theorem. It says that the remainder when a polynomial $\displaystyle f(x)$ is divided by $\displaystyle x-a$ or $\displaystyle a-x$ is $\displaystyle f(a)$.
• Dec 13th 2011, 04:50 AM
BookEnquiry
Re: Polynomial division
$\displaystyle 4x^2-3x-2$ is divided by$\displaystyle -x+1$

let $\displaystyle ax+b$ be the quotient, then $\displaystyle (ax+b)(-x+1)=4x^2-3x-2$

$\displaystyle \begin{array}{rr|l}a&b&\times\\\hline-a&-b&-1\\a&b&1\end{array}$

$\displaystyle \begin{array}{c}4=-a\\-3=-b+a\\find\ a,\ b\end{array}$

$\displaystyle remainder=-2-b$
• Dec 13th 2011, 05:26 PM
Soroban
Re: Polynomial division
Hello, luke11121!

Quote:

$\displaystyle (4x^2 - 3x - 2) \,\div\,(1 - x)$

The long division should look like this:

. . $\displaystyle \begin{array}{cccccccc} &&& - & 4x & - & 1 \\ && -- & -- & -- & -- & -- \\ -x + 1 & | & 4x^2 & - & 3x & - & 2 \\ && 4x^2 &-& 4x \\ && -- & -- & -- \\ &&&& x & - & 2 \\ &&&& x & - & 1 \\ &&&& -- & --& -- \\ &&&&& - & 1 \end{array}$

$\displaystyle 4x^2 - 3x + 2 \;=\;(-x+1)(-4x-1) + (-1)$
m . . . . . . . . . . . $\displaystyle \text{divisor}\;\;\text{quotient} \;\;\text{remainder}$

• Dec 14th 2011, 11:19 AM
Thinx
Re: Polynomial division
Obviously the problem you have give relates to quadratic function which can have two distinct real roots and multiplying them gives: (x-a)(x-b)=ax^2+bx+c (identity).

So if you you have been given one of the factors suppose (x-a) is already given:

Therefore,

=> (x-b)=ax^2/x-b+bx/x-b+c/x-b (where common denominator (x-b) ).

So its comparabale to arithematic. Compare above identity to below equation:

if

x*y=a

find y

=>y=a/x

So thats the idea.

Hope i helped.