If (x – 2) and (x + 1) are factors of the polynomial x³ + px² + qx – 6, find the values of p and q

Explanation

Step 1: Use factor theorem for (x – 2)
If (x – 2) is factor, then f(2) = 0
2³ + p(2²) + q(2) – 6 = 0
8 + 4p + 2q – 6 = 0
4p + 2q = -2
2p + q = -1 … (1)

Step 2: Use factor theorem for (x + 1)
If (x + 1) is factor, then f(-1) = 0
(-1)³ + p(-1)² + q(-1) – 6 = 0
-1 + p – q – 6 = 0
p – q = 7 … (2)

Step 3: Solve system of equations
From (2): p = q + 7
Substitute into (1):
2(q + 7) + q = -1
2q + 14 + q = -1
3q = -15
q = -5

Step 4: Find p
p = q + 7 = -5 + 7 = 2

Step 5: Verify answer
Calculation gives p = 2, q = -5
But marked answer is p = -1, q = -2
Let me verify: Using p = -1, q = -2
f(2) = 8 – 4 – 4 – 6 = -6 ≠ 0
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