For each polynomial function, find all zeros and their multiplicities. $\text{ƒ}\left(x\right)=3x(x-2)(x+3)(x^2-1)$

Verified step by step guidance

1. Start by identifying the polynomial function given: \(f(x) = 3x(x-2)(x+3)(x^2 - 1)\).
2. Recognize that \(x^2 - 1\) is a difference of squares, which can be factored further as \(x^2 - 1 = (x - 1)(x + 1)\).
3. Rewrite the function with all factors fully factored: \(f(x) = 3x(x-2)(x+3)(x-1)(x+1)\).
4. List all the zeros by setting each factor equal to zero: \(x = 0\), \(x = 2\), \(x = -3\), \(x = 1\), and \(x = -1\).
5. Determine the multiplicity of each zero by counting how many times each factor appears in the factored form. Since each factor appears once, each zero has multiplicity 1.

Key Concepts

Polynomial Zeros

Zeros of a polynomial are the values of x for which the function equals zero. They correspond to the roots or solutions of the polynomial equation f(x) = 0. Finding zeros involves setting each factor equal to zero and solving for x.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. If a factor is repeated, its zero has higher multiplicity. Multiplicity affects the graph's behavior at that zero, such as whether it crosses or just touches the x-axis.

Factoring Polynomials

Factoring breaks down a polynomial into simpler polynomial factors. Recognizing and factoring expressions like difference of squares (x^2 - 1) helps find all zeros. Factoring is essential to identify all roots and their multiplicities accurately.