End Behavior and Multiplicity Practice

Front

What is the definition of multiplicity in the context of polynomial functions?

Back

Multiplicity refers to the number of times a particular root (or zero) appears in a polynomial function. For example, if (x - r) is a factor of the polynomial, and it appears k times, then the root r has a multiplicity of k.

Front

How does the multiplicity of a zero affect the graph of a polynomial function?

Back

If a zero has an odd multiplicity, the graph will cross the x-axis at that zero. If a zero has an even multiplicity, the graph will touch the x-axis but not cross it.

Front

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

Back

The end behavior will rise to positive infinity on both ends of the graph.

Front

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

Back

The end behavior will fall to negative infinity on the right and rise to positive infinity on the left.

Front

Given the polynomial f(x) = (x - 1)(x + 4)^2, what is the multiplicity of the zero at x = 1?

Back

The multiplicity of the zero at x = 1 is 1.

Front

Given the polynomial f(x) = (x - 1)(x + 4)^2, what is the multiplicity of the zero at x = -4?

Back

The multiplicity of the zero at x = -4 is 2.

Front

What is the significance of the leading coefficient in determining the end behavior of a polynomial function?

Back

The leading coefficient determines the direction in which the graph rises or falls as x approaches positive or negative infinity.

Front

How can you determine the number of real zeros of a polynomial function?

Back

The number of real zeros can be determined by analyzing the degree of the polynomial and its factors, as well as using the Fundamental Theorem of Algebra.

Front

What is the Fundamental Theorem of Algebra?

Back

The Fundamental Theorem of Algebra states that every non-constant polynomial function of degree n has exactly n roots in the complex number system, counting multiplicities.

Front

If a polynomial has a degree of 4, what is the maximum number of real zeros it can have?

Back

The maximum number of real zeros a polynomial of degree 4 can have is 4.