Zeroes and Multiplicity of Polynomial Equations

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

How do you determine the degree of a polynomial like f(x) = x^3(x + 3)^2(x - 5)?

Back

Add the exponents of each factor: 3 (from x^3) + 2 (from (x + 3)^2) + 1 (from (x - 5)) = 6.

Front

What are the zeros of a polynomial?

Back

The zeros of a polynomial are the values of x that make the polynomial equal to zero.

Front

What does multiplicity mean in relation to polynomial zeros?

Back

Multiplicity refers to the number of times a particular zero is repeated in a polynomial.

Front

If a polynomial has a zero of -5 with a multiplicity of 3, how is it represented?

Back

It is represented as (x + 5)^3.

Front

What is the polynomial for the zeros -5 (multiplicity 3), 9 (multiplicity 2), -2, and 4?

Back

The polynomial is (x + 5)^3(x - 9)^2(x + 2)(x - 4).

Front

How do you find the zeros of the polynomial f(x) = (x + 2)(x - 3)^2(x - 4)?

Back

Set each factor equal to zero: x + 2 = 0, x - 3 = 0, x - 4 = 0. The zeros are -2, 3 (multiplicity 2), and 4.

Front

What is the significance of the degree of a polynomial?

Back

The degree of a polynomial indicates the maximum number of zeros it can have and the behavior of the graph at the ends.

Front

How do you find the degree of a polynomial expressed in factored form?

Back

Sum the exponents of all factors in the polynomial.

Front

What is the degree of the polynomial f(x) = x^3(x + 3)^2(x - 5)?

Back

The degree is 6.