Polynomials (other)

Beyond basic operations and factorization, polynomials have many interesting properties and applications. These include understanding the Remainder Theorem, Factor Theorem, and the behavior of polynomials under various transformations. Polynomials can also be used to model real-world situations and solve equations of higher degree.

Important properties

• The Remainder Theorem: When a polynomial $f(x)$ is divided by $x - a$, the remainder is $f(a)$.

• The Factor Theorem: $x - a$ is a factor of $f(x)$ if and only if $f(a) = 0$.

• A polynomial of degree $n$ has at most $n$ real roots.

• Polynomials are continuous and smooth (no jumps or sharp corners).

• Polynomials can be added, subtracted, multiplied, and (with some restrictions) divided.