Find a and b so that the polynomial x3– 10x2 + ax + b is exactly divisible by the polynomials (x – 1) and (x – 2).
Let p(x) = x3– 10x2 + ax + b Since p(x) is exactly divisible by the polynomials (x – 1) and (x – 2). ∴ By putting x = 1, we obtain (1)3 – 10(1)2 + a(1) + b = 0 ⇒ a + b = 9 And by putting x = 2, we obtain (2)3 – 10(2)2 + a(2) + b = 0 8 – 40 + 2a + b = 0 ⇒ 2a + b = 32 Subtracting (i) from (ii), we have a = 23 From (i), we have 23 + b = 9 = b = -14 Hence, the values of a and b are a = 23 and b = -14
