If x - 1 and x + 1 are both factors of the equation x\(^3\) + px\(^2\) + qx + 6 = 0, evaluate p and q
A
-6, -1
B
6, 1
C
1, -1
D
6, -6
Verified Correct Answer: A.
Step-by-Step Mathematical Solution
\(x^{3} + px^{2} + qx + 6 = 0\)
f(x - 1) = 0; f(1) = 0.
\(1^{3} + p(1^{2}) + q(1) + 6 = 0 \implies p + q = -7 ... (1)\)
f(x + 1) = 0; f(-1) = 0.
\((-1)^{3} + p(-1^{2}) + q(-1) + 6 = 0 \implies p - q = -5 ... (2)\)
Subtract (2) from (1).
\(2q = -2 \implies q = -1\)
\(p - (-1) = -5 \implies p = -5 - 1 = -6\)
\((p, q) = (-6, -1)\)
Source: Myschool Past Questions Archive (Record #30885)