For each polynomial function, find all zeros and their multiplicities. ƒ(x)=3x(x-2)(x+3)(x^2-1)

Verified step by step guidance

1. Identify the factors of the polynomial function \( f(x) = 3x(x-2)(x+3)(x^2-1) \).
2. Recognize that \( x^2 - 1 \) is a difference of squares, which can be factored further as \( (x-1)(x+1) \).
3. Rewrite the polynomial as \( f(x) = 3x(x-2)(x+3)(x-1)(x+1) \).
4. Set each factor equal to zero to find the zeros: \( x = 0, x = 2, x = -3, x = 1, x = -1 \).
5. Determine the multiplicity of each zero by examining the exponents of the factors. Since each factor appears only once, each zero has a multiplicity of 1.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 'n' is a non-negative integer and 'a_n' is not zero. Understanding polynomial functions is crucial for analyzing their behavior, including finding zeros.

Zeros of a Polynomial

The zeros of a polynomial function are the values of 'x' for which the function equals zero, i.e., f(x) = 0. These points are also known as roots and can be found by factoring the polynomial or using the quadratic formula for second-degree polynomials. Identifying zeros is essential for understanding the function's graph and its intersections with the x-axis.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular root appears in the factorization of a polynomial. If a polynomial can be expressed as (x - r)^m, where 'r' is a root and 'm' is the multiplicity, then 'm' indicates how many times 'r' is a solution. The multiplicity affects the graph's behavior at the zero, such as whether the graph crosses or touches the x-axis.