If (x + 1) and (x – 2) are factors of the polynomial (g(x) = x^{4} + ax^{3} + bx^{2} – 16x – 12), find the values of a and b.
Explanation
If (x + 1) and (x – 2) are factors of the polynomial (g(x)), then (g(-1) & g(2) = 0)
i.e. (g(-1) = (-1)^{4} + a(-1)^{3} + b(-1)^{2} – 16(-1) – 12 = 0)
(1 – a + b + 16 – 12 = 0 implies a – b = 5 … (1))
(g(2) = (2)^{4} + a(2)^{3} + b(2)^{2} – 16(2) – 12 = 0)
(16 + 8a + 4b – 32 – 12 = 0 implies 8a + 4b = 28 … (2))
From (1), (a = 5 + b).
(2) becomes : (8(5 + b) + 4b = 28 implies 40 + 8b + 4b = 28)
(40 + 12b = 28 implies 12b = -12)
(b = -1)
(a = 5 + b = 5 + (-1) = 4)
((a, b) = (4, -1))