Thread: A general case of polynomial long division?

1. A general case of polynomial long division?

I can't remember how to do polynomial long division (I know, its easy, I shouldn't have forgetten) and I'm having trouble understanding the online explainiations. Can somebody take me through the steps of how to divide: $\displaystyle (x-a)^2$ into $\displaystyle x^4 - 4a^3x+3a^4$? So if we let:

$\displaystyle f(x) = x^4 - 4a^3x+3a^4$

and

$\displaystyle h(x) = (x-a)^2 = (x-a)(x-a) = x^2 - 2ax +a^2$

Can somebody show me how to use polynomial long division on:

$\displaystyle g(x) = \frac{x^4 - 4a^3x+3a^4}{(x-a)^2} = \frac{x^4 - 4a^3x+3a^4}{(x-a)(x-a)} = \frac{x^4 - 4a^3x+3a^4}{x^2-2ax+a^2}$

??

2. You have done the right thing by expanding the demoninator in g(x).

Long division is pretty tricky in Latex so I suggest read this.

Polynomial Long Division

3. Hello, mfetch22!

I'll give it a try . . .

$\displaystyle (x^4 - 4a^3x+3a^4) \div (x-a)^2$

$\displaystyle \begin{array}{cccccccccccc} &&&&&& x^2 & + & 2ax & + & 3a^2 \\ && --&--&--&--&--&--&--&-- &--\\ x^2-2ax + a^2 & | & x^4&&&&& - & 4a^3x &-& 3a^4 \\ && x^4 & - & 2ax^3 & + & a^2x^2 \\ && --&--&--&--&-- \\ &&&& 2ax^3 &-& a^2x^2 &-& 4a^3x \\ &&&& 2ax^3 &-& 4a^2x^2 &+& 2a^3x \\ &&&& --&--&--&--&-- \\ &&&&&& 3a^2x^2 &-& 6a^3x &+& 3a^4 \\ &&&&&& 3a^2x^2 &-& 6a^3x &+& 3a^4 \\ &&&&&& --&--&--&--&-- \end{array}$