The polynomial 2x^3–ax^2+5bx+4b has a factor x–2 and, when divided by x+1, a remainder of –15 is obtained. Find the values of a and b. With these values of a and b, factorize the polynomial completely.
If x - 2 is a factor....then 2 is a zero
Thus
2(2)^3 - a(2)^2 + 5b(2) + 4b = 0
16 - 4a + 10b + 4b = 0
-4a + 14b = -16
2a - 7b = 8 (1)
And
2(-1)^3 - a(-1)^2 + 5b(-1) + 4b = -15
-2 - a - 5b + 4b = -15
-a - b = -13 ⇒ -2a - 2b = -26 (2)
Add (1) and (2) and we have
-9b = -18 ⇒ b = 2
So
-a - 2 = -13
-a = -11
a = 11
So....the polynomial is
2x^3 - 11x^2 + 5(2)x + 4(2)
2x^3 - 11x^2 + 10x + 8
And we can write
2x^3 [ - 4x^2 - 7x^2 ] + 10x + 8
[2x^3 - 4x^2] - [7x^2 - 10x - 8 ]
2x^2 (x - 2) - [ 7x^2 - 10x - 8 ]
2x^2 (x - 2) - [ ( 7x + 4) (x - 2) ]
(x - 2) (2x^2 - 7x - 4)
(x - 2) (2x + 1) ( x - 4)