polynomials

[IB231997]

i could not solve this sum..i

if you could help..

when the polynomial p(x)= x^4 + ax + 2 is divided by x^2 + 1 the remainder is 2x + 3.

find the value of a.

any kind of help would be appreciated!

[Alex .]

You may find these videos helpful: https://www.youtube.com/user/AlRichards314/videos?query=polynomial+division

Also maths tutorials: https://www.youtube.com/user/bullcleo1

[Fermat]

OK so first of,do the polynomial long division. You should get the remainder (ax+3), I assume you know how to do this already so I wont go through it step by step. So now that you know that the remainder is (ax+3) and that it is equal to 2x+3, you can just compare the coefficients of x. Because if those two expressions are going to be equal, thier x coefficients must be equal and the constant terms has to be equal.

So you get that (ax+3) = (2x+3) <=> a = 2

Edited September 6, 2012 by Fermat

[macrofire]

Here's an even simpler way:

Take your polynomial: x^4 + ax + 2. Subtract the remainder: 2x + 3

You will get x^4 + (a-2)x -1. Because the remainder is taken from this polynomial, you know that this sum has the factor x^2 + 1.

Reorder your polynomial to get x^4 -1 + (a-2)x. Note that x^4-1 = (x^2+1)(x^2-1). We know that if the polynomial is divisible by x^2 +1, then its components also have to be divisible by x^2 +1. We see that x^4 -1 is divisible by x^2+1, so that means that (a-2)x is also divisible by x^2+1. This is only possible if (a-2)x = 0, (try and figure out why). Because we are primarily interested in the coefficients, then a-2=0, or a=2.

Realize that you can do these questions using analysis and simple manipulation, instead of long division.