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 the leading coefficient and $$n$$ is the degree.

Front

What is the leading term of a polynomial?

Back

The leading term of a polynomial is the term with the highest degree. It determines the end behavior of the polynomial function.

Front

How does the leading coefficient affect the graph of a polynomial function?

Back

The leading coefficient determines the direction of the graph as x approaches positive or negative infinity. A positive leading coefficient means the graph rises to the right, while a negative leading coefficient means it falls to the right.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. It indicates the number of roots and the end behavior of the graph.

Front

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

Back

The end behavior will rise on both ends, meaning as $$x \rightarrow \infty$$, $$f(x) \rightarrow \infty$$ and as $$x \rightarrow -\infty$$, $$f(x) \rightarrow \infty$$.

Front

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

Back

The end behavior will rise to the right and fall to the left, meaning as $$x \rightarrow \infty$$, $$f(x) \rightarrow \infty$$ and as $$x \rightarrow -\infty$$, $$f(x) \rightarrow -\infty$$.

Front

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

Back

The end behavior will fall on both ends, meaning as $$x \rightarrow \infty$$, $$f(x) \rightarrow -\infty$$ and as $$x \rightarrow -\infty$$, $$f(x) \rightarrow -\infty$$.

Front

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

Back

The end behavior will fall to the right and rise to the left, meaning as $$x \rightarrow \infty$$, $$f(x) \rightarrow -\infty$$ and as $$x \rightarrow -\infty$$, $$f(x) \rightarrow \infty$$.

Front

What is the significance of the roots of a polynomial function?

Back

The roots (or zeros) of a polynomial function are the values of x for which the function equals zero. They indicate where the graph intersects the x-axis.

Front

What is a multiplicity of a root in a polynomial function?

Back

The multiplicity of a root refers to the number of times a particular root appears in the factorization of the polynomial. It affects the shape of the graph at that root.