Get zeroes of a polynomial when some are known | Example-2

Finding Values of a and b for Common Roots

Given:
g(x) = x³ + 2x² + a
The polynomial P(x) is given by: x⁵ - x⁴ - 4x³ + 3x² + 3x - b.

Assume that all the zeroes of g(x) are also zeroes of P(x).
Assume:
P(x) = g(x) × Q(x)
     = (x³ + 2x² + a)(x² + px + q)

Now expand:
(x³ + 2x² + a)(x² + px + q)
= x³(x² + px + q) + 2x²(x² + px + q) + a(x² + px + q)
= x⁵ + px⁴ + qx³
  + 2x⁴ + 2px³ + 2qx²
  + ax² + apx + aq

Combine like terms:
= x⁵ + (p + 2)x⁴ + (q + 2p)x³ + (2q + a)x² + apx + aq

Compare with:
P(x) = x⁵ - x⁴ - 4x³ + 3x² + 3x - b

Matching coefficients:
1. p + 2 = -1       → p = -3
2. q + 2p = -4      → q + 2(-3) = -4 → q = 2
3. 2q + a = 3       → 2(2) + a = 3   → a = -1
4. ap = 3           → (-1)(-3) = 3   → OK
5. aq = -b          → (-1)(2) = -b   → b = 2

Final Answer:
a = -1
b = 2