Finer grained standards that are part of this one
Solutions of polynomial equations may be real, imaginary, or a combination of real and imaginary.
Imaginary solutions occur in conjugate pairs.
Given a polynomial function 𝑓(𝑥), the following statements are equivalent for any real number 𝑘, such that 𝑓(𝑘) = 0:
Polynomial equations may have fewer distinct roots than the order of the polynomial. In these situations, a root may have “multiplicity.” For instance, the polynomial equation 𝑦 = 𝑥³ − 6𝑥² + 9𝑥 has two identical factors, (𝑥 − 3), and one other factor, 𝑥. This polynomial equation has two distinct, real roots, one with a multiplicity of 2.
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
The Fundamental Theorem of Algebra states that, including complex and repeated solutions, an 𝑛ᵗʰ degree polynomial equation has exactly 𝑛 roots (solutions).