Roots & Multiplicity

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. It is also known as a root.

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 - r) appears k times, then r has a multiplicity of k.

Front

What effect does an odd multiplicity have on the graph of a polynomial at its zero?

Back

An odd multiplicity means the graph will cross the x-axis at that zero.

Front

What effect does an even multiplicity have on the graph of a polynomial at its zero?

Back

An even multiplicity means the graph will touch the x-axis and turn around at that zero.

Front

Given the polynomial f(x) = (x + 5)^3(x - 9)^2(x + 2)(x - 4), list its zeros.

Back

The zeros are -5 (multiplicity 3), 9 (multiplicity 2), -2 (multiplicity 1), and 4 (multiplicity 1).

Front

How do you determine the multiplicity of a zero from a polynomial expression?

Back

The multiplicity of a zero can be determined by the exponent of the factor associated with that zero in the polynomial.

Front

What are the zeros of the polynomial y = 3(x + 4)(x + 1)(x - 3)?

Back

The zeros are -4, -1, and 3.

Front

If a polynomial has a zero at x = -3 with multiplicity 2, what happens at this zero?

Back

The graph will touch the x-axis at x = -3 and turn around, not crossing it.

Front

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

Back

The leading coefficient determines the end behavior of the polynomial graph. If positive, the graph rises to the right; if negative, it falls to the right.

Front

How can you find the polynomial given its zeros?

Back

To find the polynomial, multiply the factors corresponding to each zero, raised to their respective multiplicities.