Exercises: Know the Fundamental Theorem of Algebra

For quadratic problems, show the discriminant. For root-counting, include multiplicity.

Grade 9·20 problems·~40 min·Common Core Math - HS Number and Quantity·standard·hsn-cn-c-9

Work through problems with immediate feedback

Warm-Up: Review What You Know

The polynomial $p(x) = x^2 + 1$ has no real roots. Does the Fundamental Theorem of Algebra apply to $p(x)$ ?

No — the FTA only applies to polynomials with complex coefficients, not real-coefficient polynomials like $p(x)$ .

Yes — the FTA applies, but only if $p(x)$ has at least one real root.

Yes — the FTA guarantees $p(x)$ has exactly $2$ roots in the complex numbers. They are $i$ and $-i$ .

No — $p(x)$ has degree $2$ but no real roots, so the FTA fails here.