Find the value of p and q so that x4+px3+2x2-3x+q is divisible by (x2-1).
Let, f (x) = x4+px3+2x2-3x+q be the given polynomial.
And, let g (x) = (x2 – 1) = (x – 1) (x + 1)
Clearly,
(x – 1) and (x + 1) are factors of g (x)
Given, g (x) is a factor of f (x)
(x – 1) and (x + 1) are factors of f (x)
From factor theorem
If (x – 1) and (x + 1) are factors of f (x) then f (1) = 0 and f (-1) = 0 respectively.
f (1) = 0
(1)4 + p (1)3 + 2 (1)2 – 3 (1) + q = 0
1 + p + 2 – 3 + q = 0
p + q = 0 (i)
Similarly,
f (-1) = 0
(-1)4 + p (-1)3 + 2 (-1)2 - 3 (-1) + q = 0
1 – p + 2 + 3 + q = 0
q – p + 6 = 0 (ii)
Adding (i) and (ii), we get
p + q + q – p + 6 = 0
2q + 6 = 0
2q = - 6
q = -3
Putting value of q in (i), we get
p – 3 = 0
p = 3
Hence, x2 – 1 is divisible by f (x) when p = 3 and q = - 3.