Additional Mathematics – Factors of polynomials | e-Consult
Factors of polynomials (1 questions)
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First, look for a rational root. Test x = ½:
4(½)³ − 12(½)² + 9(½) − 2 = 4·1/8 − 12·1/4 + 4.5 − 2 = 0.5 − 3 + 4.5 − 2 = 0.
Hence x = ½ is a root, so (x − ½) is a factor.
Divide the cubic by (x − ½):
| 4x³ − 12x² + 9x − 2 ÷ (x − ½) = 4x² − 10x + 4 |
Now solve the quadratic 4x² − 10x + 4 = 0:
Δ = (‑10)² − 4·4·4 = 100 − 64 = 36.
Thus x = [10 ± √36]/(2·4) = [10 ± 6]/8.
- x = (10 + 6)/8 = 16/8 = 2
- x = (10 ‑ 6)/8 = 4/8 = ½
We already have x = ½ from the linear factor; it appears twice (a double root).
Therefore the solutions are x = 2 and x = ½ (double root).