The polynomial P (x) = ax³-11x+bx+2, has a factor of x-2 and leaves a reminder of - 12 when divided by x+1. Find the values of a and b

a = - 5 and b = 30

Step-by-step explanation:

Using the Remainder theorem

If a polynomial p (x) is divided by (x - h) then

p (h) = Remainder

If (x - h) is a factor of p (x) then p (h) = 0

Given (x - 2) is a factor the p (2) = 0

Given (x + 1) leaves a remainder of - 12 then p ( - 1) = - 12

Hence

p (2) = a (2) ³ - 11 (2) + b (2) + 2 = 0, that is

8a - 22 + 2b + 2 = 0

8a + 2b - 20 = 0 (add 20 to both sides)

8a + 2b = 20 → (1)

and

p ( - 1) = a ( - 1) ³ - 11 ( - 1) + b ( - 1) + 2 = - 12, that is

- a + 11 - b + 2 = - 12

- a - b + 13 = - 12 (subtract 13 from both sides)

- a - b = - 25 (multiply through by - 1)

a + b = 25 (subtract b from both sides)

a = 25 - b → (2)

Substitute a = 25 - b into (1)

8 (25 - b) + 2b = 20

200 - 8b + 2b = 20

200 - 6b = 20 (subtract 200 from both sides)

- 6b = - 180 (divide both sides by - 6)

b = 30

Substitute b = 30 into (2)

a = 25 - 30 = - 5