Question

show that fundamental theorem of algebra is true for 2x^2-12x+18=0 how many rational and non rational solutions

Answer 1

The Fundamental Theorem of Algebra says a degree-n polynomial (with complex coefficients) has exactly n complex roots counting multiplicity. Here the polynomial is degree 2, so it must have two complex roots counted with multiplicity.

Compute or factor the quadratic:
2x^2 − 12x + 18 = 2(x^2 − 6x + 9) = 2(x − 3)^2.

Thus the only root is x = 3, with multiplicity 2. Equivalently, the quadratic formula gives discriminant Δ = (−12)^2 − 4·2·18 = 144 − 144 = 0, so a repeated root x = 12/(4) = 3.

Therefore:
- Number of complex roots counting multiplicity: 2 (both equal to 3), so the Fundamental Theorem of Algebra holds.
- Number of distinct rational roots: 1 (x = 3).
- Number of rational roots counting multiplicity: 2 (both are 3).
- Number of non‑rational (irrational or nonreal) roots: 0.