Question

Find the values of a and b, so that polynomial x^3 + 10 X^2 + ax + b is exactly divisible by x - 1 as well as X - 2

Answer 1

polynomial is divisible by x-1 and x-2 => 1,2 are roots of p(x)= x^3+10x^2+ax+b
So equate p(1)=0
and. p(2)=0 to get 2 eqns in a & b.
Solve them to get a and b.

Answer 2

Given : f(x) = x^3 + 10x^2 + ax + b.

(i)

Given that x - 1 is a factor of f(x).

We know that when (x - a) is a factor of f(x), then f(a) = 0.

⇒ f(1) = (1)^3 + 10(1)^2 + a(1) + b

⇒ 0 = 1 + 10 + a + b

⇒ a + b = -11    

(ii)

Given that x - 2 is also a factor of f(x)

⇒ f(2) = (2)^3 + 10(2)^2 + a(2) + b

⇒ 0 = 8 + 40 + 2a + b

⇒ 2a + b + 48 = 0

⇒ 2a + b = -48.

On solving (1) & (2), we get

⇒ a + b = -11

⇒ 2a + b = -48

   ------------------

    -a = 37

     a = -37.

Substitute a = 37 in (2), we get

⇒ 2a + b = -48

⇒ 2(-37) + b = - 48

⇒ -74 + b = -48

⇒ b = -48 + 74

⇒ b = 26.

Therefore:

⇒ The value of a = -37.

⇒ The value of b = 26.

Hope it helps!