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Core 2 A-level: remainder theorem HELP?!

When divided by x+3, the polynomial 2x³+x²+ax+1 leaves a remainder of -53. Find the value of constant a.

When divided by x+1, the polynomial ax³+bx²-13x+6 leaves a remainder of 18. Given that 2x-1 is a factor of the polynomial, find the values of a and b.


I have no idea what to do when all of the coefficients aren't there, could anyone help please? thanks! :s-smilie:
The remainder theorem states that if ff is a polynomial, then
f(x)xc has remainder R    f(c)=R\displaystyle\frac{f(x)}{x-c} \text{\ has\ remainder\ }R\iff f(c)=R .

So you must have that 2(3)3+(3)2+a(3)+1=532(-3)^3+(-3)^2+a(-3)+1=-53 for your first one. It's now easy to solve for aa .

Can you do the other?
Reply 2
Thank you! :biggrin:

Okay, I did that and got a as 3, is that right?

Erm, I am not sure how to do the other one :frown:
Just do the same thing! Substitute a suitable x into the polynomial and equate it to something (found using the remainder theorem).

Oops, sorry. Didn't read. A better formulatin of the remainder theorem is
f(x)dxc has remainder R    f(cd)=R\displaystyle\frac{f(x)}{dx-c} \text{\ has\ remainder\ }R\iff f\left( \frac cd\right) =R
(edited 13 years ago)

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