find the values os a and b so that the polynomial x3+10x2+ax+b is exactly divisible by (x-1) and (x+2)

a = 7 and b = - 18

Step-by-step explanation:

If (x - 1) is a factor of f (x) then x = 1 is a root and f (1) = 0, thus

f (1) = 1³ + 10 (1) ² + a + b = 0 and

1 + 10 + a + b = 0 ⇒ a + b = - 11 → (1)

Similarly if

(x + 2) is a factor then x = - 2 is a root and f ( - 2) = 0

f ( - 2) = ( - 2) ³ + 10 ( - 2) ² - 2a + b = 0, thus

- 8 + 40 - 2a + b = 0 ⇒ - 2a + b = - 32 → (2)

Solve (1) and (2) simultaneously

Subtract (1) from (2)

- 3a = - 21 (divide both sides by 3)

a = 7

Substitute a = 7 in (1)

7 + b = - 11 ⇒ b = - 11 - 7 = - 18

The required polynomial is

f (x) = x³ + 10x² + 7x - 18