Polynomials Question (1 Viewer)

[Riviet]

How would I find the remainder if I have a polynomial of high degree such as p(x)=x⁴⁵+1 divided by an unfactorisable quadratic like x²+4?

Any help is greatly appreciated, thanks.

 

[Raginsheep]

I get a massive coefficient but...

P(x)=A(x)Q(x)+R(x)
= (x²+4)Q(x)+R(x)
Since A(x) is deg 2, R(x) is < deg 2
Thus,
P(x)=(x²+4)Q(x)+(ax+b)

sub in x=2i and x=-2i and solve for a and b

 

[Riviet]

Ah of course, totally forgot about complex numbers.

Thanks Raginsheep.

edit: whoah, that's nuts buchanan!

 

[buchanan]

Or you could just divide by inspection:

x⁴⁵+1=(x²+4)(x⁴³-4x⁴¹+4²x³⁹-4³x³⁷+...-4²¹x)+4²²x+1

So the remainder is 4²²x+1

Edit: It's not as hard as it looks. It's just a simple pattern.

 

[buchanan]

A mathematician, like a painter or poet, is a maker of patterns. - G.H Hardy

 

[DraconisV]

Well said