[falafels]

Any help would be much appreciated...I just can't work out how to go about setting out this question as both long division and remainder theorem.

8x³+2x²+5
2x²+2

Set be set help would really help! Thanks!

[Christophicus]

[quote=falafels]Any help would be much appreciated...I just can't work out how to go about setting out this question as both long division and remainder theorem.

8x³+2x²+5
2x²+2


[8x³+2x²+5]/[2x²+2]
= (8x(x²+1)-8x+2x²+5)/2(x²+1)
= 8x(x²+1)/2(x²+1) + (2x²-8x+5)/2(x²+1)
= 4x + (2x²-8x+5)/2(x²+1)
= 4x + (2(x²+1)-8x+3)/2(x²+1)
= 4x+1 + 3-8x/2(x²+1)

[falafels]

Thanks, wow that's amazing...how did you get the common factor in the first step? And what happened to the "-8x" between the first and second step?

[Christophicus]

Originally Posted by falafels

Thanks, wow that's amazing...how did you get the common factor in the first step?

well you can take out (x²+1) on the numerator and the denominator.
For the two methods you've described, do long division with +0x as that'll help.

[Christophicus]

For the remainder theorem
8x³+2x²+5 = (2x²+2)(Ax+B) + Cx + D
8x³+2x²+5 = 2Ax³+2Bx²+2Ax+2B+Cx+D
8x³+2x²+5 = 2Ax³+2Bx²+(2A+C)x+2B+D

2A = 8 => A = 4
2B = 2 => B = 1
2A+C = 0 => C = -2A = -8
2B+D = 5 => D = 5-2B = 5-2 = 3

Therefore;
(8x³+2x²+5)/(2x²+2) = 4x+1 -8x+3/(2x²+2)

[falafels]

Thanks, but how about when the denominator can't be factorised..like:

4x³-5x²+3x-14
x²+2x-1

[Christophicus]

Originally Posted by falafels

Thanks, but how about when the denominator can't be factorised..like:

4x³-5x²+3x-14
x²+2x-1

4x³-5x²-14 = (Ax+B)(x²+2x-1) + Cx + D
4x³-5x²-14 = Ax³+2Ax²-Ax+Bx²+2Bx-B+Cx+D
4x³-5x²-14 = Ax³+(2A+B)x²+(2B+C-A)x-B+D

Comparing coefficients.
4 = A
-5 = 2A+B => B = -5-2A = -5-8 = -13
2B+C-A = 0 (no x term) => C = A-2B = 4-(-26) = 30
-B+D = -14 => D = -14+B = -14-13 = -27

(4x³-5x²-14)/(x²+2x-1) = 4x-13 + (30x-27)/(x²+2x-1)

[ahmadz5]

It dont get it! Could someone help explain this???

[Dekota]

Originally Posted by ahmadz5

It dont get it! Could someone help explain this???

You don't need to know this because you don't do C3.

[Christophicus]

Originally Posted by ahmadz5

It dont get it! Could someone help explain this???

Which part don't you get?
I'd be happy to explain it to you.