Algebra 2: Fundamental Theorem of Algebra and Graphing Polynomials

The video covers the fundamental theorem of algebra, explaining that every polynomial equation with a degree greater than zero has at least one root in the set of complex numbers. It discusses the relationship between the degree of a polynomial and its roots, including examples of polynomial equations and their roots with multiplicities. The video also provides a detailed guide on graphing higher order polynomials, outlining steps such as determining end behavior, finding real zeros, and plotting graphs using a graphing calculator.

What does the Fundamental Theorem of Algebra state about the roots of a polynomial?

• A polynomial of degree n has at least one root in the set of complex numbers.

• A polynomial of degree n has exactly n distinct real roots.

• A polynomial of degree n has no roots if n is even.

• A polynomial of degree n has only real roots.

How is the degree of a polynomial related to its roots?

• The degree is equal to the number of distinct real roots.

• The degree is always one less than the number of roots.

• The degree is equal to the total number of roots, including multiplicities.

• The degree is twice the number of complex roots.

What is the multiplicity of a root in a polynomial?

• The number of distinct real roots.

• The number of times a root appears in the factorization of the polynomial.

• The number of complex roots a polynomial has.

• The degree of the polynomial.

When graphing a polynomial, what does the leading coefficient determine?

• The y-intercept of the graph.

• The number of complex roots.

• The number of real roots.

• The end behavior of the graph.

What is the first step in graphing a polynomial equation?

• Finding the y-intercept.

• Determining the end behavior using leading coefficients and degrees.

• Plotting the zeros.

• Sketching the graph.

In the example of graphing a third-degree polynomial, what is the significance of the y-intercept?

• It is the highest point on the graph.

• It is the point where the graph crosses the x-axis.

• It determines the end behavior of the graph.

• It is the constant term of the polynomial.

What is the role of synthetic division in finding polynomial roots?

• It is used to graph the polynomial.

• It simplifies the process of finding roots.

• It is used to determine the degree of the polynomial.

• It helps in finding the y-intercept.

How does a coefficient less than one affect the graph of a polynomial?

• It makes the graph steeper.

• It flattens and smooths the graph.

• It changes the y-intercept.

• It increases the number of roots.

What is the effect of a multiplicity of two on a polynomial graph?

• The graph crosses the x-axis at the root.

• The graph has a vertical asymptote at the root.

• The graph touches and bounces off the x-axis at the root.

• The graph has a horizontal asymptote at the root.

Why is it important to verify polynomial graphs using a graphing calculator?

• To find the exact y-intercept.

• To ensure the graph is drawn to scale.

• To confirm the accuracy of the sketched graph.

• To determine the number of complex roots.