Question

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 ​

Answer 1

Answer:

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