4-13 Polynomial Functions with Multiplicity

Front

What are the zeros of a polynomial?

Back

The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

Front

How do you determine the multiplicity of a zero in a polynomial?

Back

The multiplicity of a zero is determined by the exponent of the factor associated with that zero in the polynomial's factored form. A zero with an exponent of 1 has a multiplicity of 1, an exponent of 2 has a multiplicity of 2, and so on.

Front

What is the significance of a zero with an even multiplicity on a graph?

Back

A zero with an even multiplicity touches the x-axis at that zero but does not cross it.

Front

What is the significance of a zero with an odd multiplicity on a graph?

Back

A zero with an odd multiplicity crosses the x-axis at that zero.

Front

Given the polynomial f(x) = x(x + 3)²(x - 2), what are the zeros and their multiplicities?

Back

The zeros are x = 0 (multiplicity of 1), x = -3 (multiplicity of 2), and x = 2 (multiplicity of 1).

Front

How can you find the number of real zeros from a polynomial's graph?

Back

The number of real zeros can be determined by counting the points where the graph intersects the x-axis.

Front

What does it mean if a polynomial has a zero at x = -4?

Back

It means that when x = -4, the polynomial evaluates to zero, indicating that (x + 4) is a factor of the polynomial.

Front

How do you identify the degree of a polynomial?

Back

The degree of a polynomial is the highest exponent of the variable in the polynomial's expression.

Front

What is the relationship between the degree of a polynomial and the number of zeros it can have?

Back

A polynomial of degree n can have at most n real zeros, counting multiplicities.

Front

What is the role of the leading coefficient in a polynomial function?

Back

The leading coefficient determines the end behavior of the polynomial function as x approaches positive or negative infinity.