Task Sets: Know the Fundamental Theorem of Algebra - Common Core Math - HS Number and Quantity

Warm-Up: Review What You Know

The Fundamental Theorem of Algebra states that every non-constant polynomial of degree $n$ has exactly how many roots in the complex numbers?

At least $n$ roots (could be more)

Exactly $n$ distinct roots, all different

Exactly $n$ roots, counting multiplicity

At most $n$ roots

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.

The polynomial $p(x) = (x - 2)^2(x + 5)$ has degree $3$ .

According to the FTA, how many roots does $p(x)$ have in the complex numbers, counting multiplicity?