Thread: Find the remainder on dividing the following polynomial:

1. Find the remainder on dividing the following polynomial:

P(x) = x^4 + (2-a)x^3 + (3-2a)x^2 - 2x + a
Divide the polynomial by x^2 + 2x + 2, and find the remainder
Hence, deduce that x^4 - 2x^2 + 2 is divisible by x^2 + 2x + 2 and factorise P(x) completely into real factors.
Thanks for any help. I started to use synthetic and algorithm division, but it just turned into a mess

2. Originally Posted by Zel
P(x) = x^4 + (2-a)x^3 + (3-2a)x^2 - 2x + a
Divide the polynomial by x^2 + 2x + 2, and find the remainder
Hence, deduce that x^4 - 2x^2 + 2 is divisible by x^2 + 2x + 2 and factorise P(x) completely into real factors.
Thanks for any help. I started to use synthetic and algorithm division, but it just turned into a mess
P(x) = x^4 + 2x^3 + 3X^2 -2x -ax^3 -2ax^2 +a
Now
(x^4 + 2x^3 + 3X^2 -2x)/(x^2 + 2x + 2) = (x^2) +(x^2 - 2x)/(x^2 + 2x + 2)
(-ax^3 -2ax^2 +a)/(x^2 + 2x + 2) = -a(x^3 + 2x^2 - 1)/(x^2 + 2x + 2)
= -a[ 2 + (x^3 - 4x - 5)/(x^2 + 2x + 2)]
Now you can find the remainder.

3. Well, long division on:

[x^4 + (2-a)x^3 + (3-2a)x^2 - 2x + a] / [x^2 + 2x + 2]

gives: x^2 - ax + 1 with remainder (2a-4)x + a - 2