Factorise x³ − 6x² + 11x − 6 completely using the Factor Theorem. State the zeroes of the polynomial.

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STEP 1 — FIND ONE ZERO BY TRIAL: Test factors of the constant term (−6): ±1, ±2, ±3, ±6. p(1) = 1 − 6 + 11 − 6 = 0. So x=1 is a zero, and (x−1) is a factor. STEP 2 — DIVIDE BY (x−1): x³ − 6x² + 11x − 6 ÷ (x−1). Using synthetic division or long division: Quotient = x² − 5x + 6. Check: (x−1)(x²−5x+6) = x³−5x²+6x−x²+5x−6 = x³−6x²+11x−6 ✓. STEP 3 — FACTORISE THE QUADRATIC: x² − 5x + 6. Split middle term: find two numbers with product = 6 and sum = −5 → (−2) and (−3). x²−5x+6 = (x−2)(x−3). COMPLETE FACTORISATION: x³ − 6x² + 11x − 6 = (x−1)(x−2)(x−3). ZEROES: Setting each factor = 0: x = 1, x = 2, x = 3. VERIFICATION: p(1) = 1−6+11−6 = 0 ✓. p(2) = 8−24+22−6 = 0 ✓. p(3) = 27−54+33−6 = 0 ✓.

Practice — Polynomials and Factorisation

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