Remainder theorem precalc

Question

f(x) = 6x^5 + 13x^4 - 30x^3 - 45x^2 + ax + b is divisible by (2x-1) and when divided by (x-1) the remainder is -40. Find the values of a and b.

Answer 1

divide the polynomial by 2x-1 and you get b+(a-28)/2 as a remainder which is equal to 0 because the polynomial is divisible by 2x-1

next divide the polynomial by x-1 and you get b+(a-56) as a remainder which is equal to -40

you must be very careful when you divide because it gets difficult when you get to the "ax+b" part of the polynomial

b+(a-28)/2=0 and b+(a-56)=-40

simplify both equations wherever possible

2b+a-28=0

a+2b=28

b+(a-56)=-40

a+b=56-40

a+b=16

a+2b=28

a+b=16

subtract

b=12

so a=4