Dec 5 Polynomial Roots and Multiplicities

Front

What is a polynomial root?

Back

A polynomial root is a value of x that makes the polynomial equal to zero. For example, in the polynomial f(x) = x^2 - 4, the roots are x = 2 and x = -2.

Front

Define multiplicity in the context of polynomial roots.

Back

Multiplicity refers to the number of times a particular root appears in a polynomial. For example, in f(x) = (x - 2)^3, the root x = 2 has a multiplicity of 3.

Front

Find the roots of the polynomial: f(x) = x^2 - 5x + 6.

Back

The roots are x = 2 and x = 3.

Front

What is the significance of a root's multiplicity?

Back

The multiplicity of a root affects the graph of the polynomial. A root with an odd multiplicity crosses the x-axis, while a root with an even multiplicity touches the x-axis but does not cross it.

Front

Given f(x) = (x + 1)(x - 3)^2, identify the roots and their multiplicities.

Back

Roots: x = -1 (multiplicity 1), x = 3 (multiplicity 2).

Front

Solve for the roots: 0 = x^3 - 6x^2 + 9x.

Back

The roots are x = 0, x = 3 (multiplicity 2).

Front

What is the root of the polynomial f(x) = x^2 + 4?

Back

There are no real roots since the polynomial does not cross the x-axis.

Front

Identify the roots of the polynomial: f(x) = x(x - 2)(x + 3).

Back

The roots are x = 0, x = 2, x = -3.

Front

What is the multiplicity of the root x = 1 in f(x) = (x - 1)^4?

Back

The multiplicity is 4.

Front

Find the roots of the polynomial: f(x) = x^2 - 1.

Back

The roots are x = 1 and x = -1.