If (x + 2) and (x – 1) are factors of the polynomial p(x) = x3 + 10x2 + mx + n then
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Given that (x + 2) and (x – 1) are factors of p(x) = x3 + 10x2 + mx + n, then p(– 2) = 0 and p(1) = 0 by the Factor Theorem.
First, solve p(– 2) = 0: (− 2)3 + 10 × (− 2)2 + m × (− 2) + n = 0 → – 8 + 40 – 2m + n = 0 → – 2m + n = – 32. (Equation 1)
Next, solve p(1) = 0: 13 + 10 × 12 + m × 1 + n = 0 → 1 + 10 + m + n = 0 → m + n = – 11. (Equation 2)
Subtract Equation 2 from Equation 1: (– 2m + n) – (m + n) = – 32 – (– 11) → – 3m = – 21 → m = 7.
Substitute m = 7 into Equation 2: 7 + n = – 11 → n = – 18.
Final Answer: m = 7, n = – 18