Zeros, Multiplicities, and Degrees of Polynomials

Front

What are the roots of the polynomial function? y = (x + 2)(x - 7)^3

Back

The roots are -2 and 7 with multiplicity 3.

Front

What is the polynomial with 3 factors of (x – 1) and 2 factors of (x + 7)?

Back

The polynomial is b(x) = (x - 1)^3(x + 7)^2.

Front

What is the equation of a polynomial with 5 factors of (x – 5), 6 factors of (x + 5), and 4 factors of (x + 4)?

Back

The polynomial is f(x) = (x - 5)^5(x + 5)^6(x + 4)^4.

Front

Define a root of a polynomial.

Back

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

Front

What is multiplicity in the context of polynomial roots?

Back

Multiplicity refers to the number of times a particular root appears in a polynomial.

Front

How do you determine the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial.

Front

What is the degree of the polynomial p(x) = 4x^3 - 2x^2 + 7?

Back

The degree of the polynomial is 3.

Front

If a polynomial has a root of x = 3 with multiplicity 2, what does this imply about the graph of the polynomial?

Back

The graph will touch the x-axis at x = 3 and turn around, indicating a repeated root.

Front

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

Back

A polynomial of degree n can have up to n roots, counting multiplicities.

Front

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

Back

The leading coefficient determines the end behavior of the polynomial's graph.