Polynomials - End Behavior and Graphs

Front

What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, 3x^2 + 2x + 1 is a polynomial.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial 4x^3 + 2x^2 + 5, the degree is 3.

Front

How does the end behavior of a polynomial function relate to its degree?

Back

The end behavior of a polynomial function is determined by its leading term. If the degree is even, both ends of the graph will go in the same direction. If the degree is odd, the ends will go in opposite directions.

Front

What is the leading coefficient of a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. For example, in 5x^4 + 3x^2 + 2, the leading coefficient is 5.

Front

Describe the end behavior of a polynomial function with a positive leading coefficient and an even degree.

Back

As x approaches -∞, f(x) approaches +∞; as x approaches +∞, f(x) approaches +∞.

Front

Describe the end behavior of a polynomial function with a negative leading coefficient and an even degree.

Back

As x approaches -∞, f(x) approaches -∞; as x approaches +∞, f(x) approaches -∞.

Front

Describe the end behavior of a polynomial function with a positive leading coefficient and an odd degree.

Back

As x approaches -∞, f(x) approaches -∞; as x approaches +∞, f(x) approaches +∞.

Front

Describe the end behavior of a polynomial function with a negative leading coefficient and an odd degree.

Back

As x approaches -∞, f(x) approaches +∞; as x approaches +∞, f(x) approaches -∞.

Front

What is the significance of the x-intercepts of a polynomial function?

Back

The x-intercepts of a polynomial function are the points where the graph crosses the x-axis. They represent the values of x for which f(x) = 0.

Front

How can you determine the number of x-intercepts a polynomial can have?

Back

A polynomial of degree n can have at most n real x-intercepts, but it may have fewer due to multiplicity or complex roots.