The expression x^3+a(x^2)+bx–5 is exactly divisible by (x–1). When the same expression is divided by (x–2) the remainder is 15. Find the value of the expression when x=–1.

Answer 1

divide the polynomial by x-1 to get x^2+(a+1)x+(a+b+1) for a quotient

you will get a+b-4 for a remainder which must equal 0

therefore you get a+b=4 for one equation

divide the polynomial by x-2 and you will get x^2+(a+2)x+(2a+b+4) for the quotient

you also get 4a+2b+3 for a remainder which must equal 15

so, 4a+2b+3=15 and 4a+2b+3=15

so, 4a+2b=12 and 2a+b=6 for the second equation

2a+b=6

a+b=4

solve this system of equations and a=2 and b=2

substituting you get x^3+2x^2+2x-5 for the expression

if x=-1, the expression equals -1+2-2-5=-6