can someone please find the remainder when the polynomial f(x)=x^4 - x^3 + 3x^2 -2x + 1 is divided by (x-2). It'd be such a huge help!!! can someone please find the remainder when the polynomial f(x)=x^4 - x^3 + 3x^2 -2x + 1 is divided by (x-2). It'd be such a huge help!!! @Mathematics

Question

anonymous · Answer

plug 2 into the polynomial.

anonymous · Answer

I need to show working of the long division, and I don't know how to do long division properly.

anonymous · Answer

oh <.< yeah thats going to require a little more work.  are you familiar with synthetic division?

agreene · Answer

http://www.wolframalpha.com/input/?i=%28x%5E4+-+x%5E3+%2B+3x%5E2+-2x+%2B+1%29++divided+by+%28x-2%29 scroll down to quotient and remainder and click show steps.

anonymous · Answer

sweeeet. thanks :)

anonymous · Answer

Notice how the remainder is 17, and f(2) is also 17.   Its a really fast shortcut to know if you have a polynomial p(x), and you want the remainder when you divide by (x-k), your answer is p(k), just plug k into the polynomial.

agreene · Answer

lol nice joe... never learnt that one!

anonymous · Answer

not many do.  its because of the division algorithm.  When you divide x^4-x^3+3x^2-2x+2 by x-2 you end up with:

$$x^4-x^3+3x^2-2x+1=(x^3+x^2+5x+8)(x-2)+R$$where R is the remainder. this can be written in a nicer way:$$f(x)=(x^3+x^2+5x+8)(x-2)+R$$plugging in 2 gives:$$f(2)=(x^3+x^2+5x+8)(2-2)+R=R$$Since 2-2= 0.  this always works.

agreene · Answer

interesting... never really thought about it. I really only did polynomial division when I was in like grade 7, and thus my algebra was horrible and I never really thought about that... elegant and easy--just how I like proofs :)