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# Dividing a polynomial by another watch

1. How to do this question?

Let f(x) = 7x^32 + 5x^22 + 3x^12 + x^2 . Find the remainder when f(x) is divided by x^2 + 1.

Is there an easier method than long division. If there isnt... how do you do this by long division as i havent learn that yet
2. (Original post by Carman3)
How to do this question?

Let f(x) = 7x^32 + 5x^22 + 3x^12 + x^2 . Find the remainder when f(x) is divided by x^2 + 1.

Is there an easier method than long division. If there isnt... how do you do this by long division as i havent learn that yet
Notice that all your terms in the polynomial are integral powers of x^2.

So, let y=x^2.

Your polynomial now becomes f(y) = 32y^16+ ....
And you want the remainder when it's divided by y+1

Doable now?

Edit: To be strictly accurate, I should say g(y) rather than f(y), since this changes the function, and we have g(x^2) =f(x)
3. If you know about complex numbers, factorize x^2+1 and use the remainder theorem.
4. (Original post by ghostwalker)
Notice that all your terms in the polynomial are integral powers of x^2.

So, let y=x^2.

Your polynomial now becomes f(y) = 32y^16+ ....
And you want the remainder when it's divided by y+1

Doable now?
So you would get 7y^16+5y^11+3y^6+y

What would you do after that... factorise it? If so, how
5. (Original post by Carman3)
So you would get 7y^16+5y^11+3y^6+y

What would you do after that... factorise it? If so, how
Remainder theorem.
6. (Original post by ghostwalker)
Notice that all your terms in the polynomial are integral powers of x^2.

So, let y=x^2.
Our posts must have crossed - much nicer suggestion than mine. PRSOM...

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