Polynomial Functions

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: $$f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$ where $$a_n$$ is not zero.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in $$f(x) = 2x^3 - 3x^2 + 4x - 10$$, the degree is 3.

Front

What does it mean for a polynomial to have an odd degree?

Back

A polynomial has an odd degree if the highest exponent of its variable is an odd number. Odd degree polynomials have opposite end behaviors: as $$x \to -\infty$$, $$f(x) \to -\infty$$ and as $$x \to \infty$$, $$f(x) \to \infty$$.

Front

What is the end behavior of a polynomial function with a positive leading coefficient and an odd degree?

Back

As $$x \to -\infty$$, $$f(x) \to -\infty$$; and as $$x \to \infty$$, $$f(x) \to \infty$$.

Front

What is the end behavior of a polynomial function with a negative leading coefficient and an odd degree?

Back

As $$x \to -\infty$$, $$f(x) \to \infty$$; and as $$x \to \infty$$, $$f(x) \to -\infty$$.

Front

What is the maximum number of turns a polynomial graph can have?

Back

The maximum number of turns a polynomial graph can have is one less than its degree. For example, a polynomial of degree 3 can have a maximum of 2 turns.

Front

What is a leading coefficient?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It determines the direction of the graph's end behavior.

Front

How do you determine the end behavior of a polynomial function?

Back

The end behavior of a polynomial function can be determined by the leading term, which is the term with the highest degree. The sign of the leading coefficient and the degree (odd/even) dictate the behavior.

Front

What is the difference between even and odd degree polynomials?

Back

Even degree polynomials have the same end behavior on both sides (both go to infinity or both go to negative infinity), while odd degree polynomials have opposite end behaviors.

Front

What is the significance of the roots of a polynomial?

Back

The roots of a polynomial are the values of $$x$$ for which the polynomial equals zero. They indicate where the graph intersects the x-axis.