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

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Start by identifying the polynomial function given: $f(x) = 3x(x-2)(x+3)(x^2 - 1)$.

Recognize that $x^2 - 1$ is a difference of squares, which can be factored further as $x^2 - 1 = (x - 1)(x + 1)$.

Rewrite the function with all factors fully factored: $f(x) = 3x(x-2)(x+3)(x-1)(x+1)$.

List all the zeros by setting each factor equal to zero: $x = 0$, $x = 2$, $x = -3$, $x = 1$, and $x = -1$.

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.

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주요 개념

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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.

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=3x(x−2)(x+3)(x2−1)ƒ(x)=3x(x-2)(x+3)(x^2-1)