Key Features of Polynomials

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, combined using addition, subtraction, and multiplication.

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.

Front

What does it mean for a polynomial to be decreasing on an interval?

Back

A polynomial is decreasing on an interval if, as the input values increase, the output values decrease.

Front

What does it mean for a polynomial to have a maximum?

Back

A polynomial has a maximum at a point if the value of the polynomial at that point is greater than the values at nearby points.

Front

What is a root of a polynomial?

Back

A root of a polynomial is a value of the variable that makes the polynomial equal to zero.

Front

What is multiplicity in relation to polynomial roots?

Back

Multiplicity refers to the number of times a particular root appears in a polynomial. A root with multiplicity greater than 1 indicates that the graph touches the x-axis at that root.

Front

How can you determine the intervals where a polynomial is increasing or decreasing?

Back

To determine the intervals where a polynomial is increasing or decreasing, find the derivative of the polynomial, set it to zero to find critical points, and test the intervals around those points.

Front

What is the significance of the leading coefficient in a polynomial?

Back

The leading coefficient determines the end behavior of the polynomial function. If it is positive, the graph rises to the right; if negative, it falls to the right.

Front

What is the difference between a local maximum and an absolute maximum?

Back

A local maximum is the highest point in a specific interval, while an absolute maximum is the highest point over the entire domain of the polynomial.

Front

What is the role of the x-intercepts in a polynomial graph?

Back

The x-intercepts of a polynomial graph represent the roots of the polynomial, where the graph crosses the x-axis.