Composition of polynomials
Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.
What is the degree of the following polynomials?
ƒ ⋅ f

Key Concepts

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial f(x) = 3x^4 + 2x^2 + 1, the degree is 4. The degree provides important information about the polynomial's behavior, including the number of roots and the end behavior of its graph.

Multiplication of Polynomials

When multiplying polynomials, the degree of the resulting polynomial is the sum of the degrees of the polynomials being multiplied. For instance, if f is an nth-degree polynomial and g is an mth-degree polynomial, then the degree of the product f ⋅ g is n + m. This principle is crucial for determining the degree of polynomial expressions resulting from operations.

Composition of Polynomials

The composition of polynomials involves substituting one polynomial into another. For example, if f(x) is a polynomial and g(x) is another, then the composition f(g(x)) results in a new polynomial. While the question focuses on multiplication, understanding composition helps clarify how polynomials interact, especially in more complex expressions.