multiplicity and polynomials

Front

What is a zero of a polynomial?

Back

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

Front

Define multiplicity in the context of polynomial roots.

Back

Multiplicity refers to the number of times a particular root (zero) appears in a polynomial. For example, if (x - 3) appears twice, then 3 has a multiplicity of 2.

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 expression.

Front

What is the significance of even multiplicity at a root?

Back

If a root has even multiplicity, the graph of the polynomial touches the x-axis at that root but does not cross it.

Front

What is the significance of odd multiplicity at a root?

Back

If a root has odd multiplicity, the graph of the polynomial crosses the x-axis at that root.

Front

Given f(x) = (x + 1)(x - 2)(x - 3), identify the zeros.

Back

The zeros are -1, 2, and 3.

Front

What is the polynomial formed by the roots -1 (multiplicity 2) and 2?

Back

The polynomial is f(x) = (x + 1)²(x - 2).

Front

How do you find the roots of a polynomial function?

Back

To find the roots, set the polynomial equal to zero and solve for x.

Front

What does it mean if a polynomial has a root of multiplicity 3?

Back

It means that the root appears three times in the factorization of the polynomial.

Front

What is the polynomial for the roots 1 (multiplicity 3) and -2 (multiplicity 2)?

Back

The polynomial is f(x) = (x - 1)³(x + 2)².