For each polynomial function, find all zeros and their multiplicities. $\text{ƒ}\left(x\right)=(x+1{)}^2(x-1{)}^3(x^2-10)$

Verified step by step guidance

1. Identify the factors of the polynomial function given: \(f(x) = (x+1)^2 (x-1)^3 (x^2 - 10)\).
2. Set each factor equal to zero to find the zeros of the function: solve \(x+1=0\), \(x-1=0\), and \(x^2 - 10=0\) separately.
3. Solve the linear equations: \(x+1=0\) gives \(x=-1\), and \(x-1=0\) gives \(x=1\). These are zeros with multiplicities corresponding to the exponents on their factors.
4. Solve the quadratic equation \(x^2 - 10 = 0\) by isolating \(x^2\): \(x^2 = 10\), then take the square root of both sides to find \(x = \pm \sqrt{10}\).
5. Determine the multiplicities of each zero based on the exponents in the original polynomial: \(x=-1\) has multiplicity 2, \(x=1\) has multiplicity 3, and \(x=\pm \sqrt{10}\) each have multiplicity 1.

Key Concepts

Polynomial Zeros

Zeros of a polynomial are the values of x for which the polynomial equals zero. Finding zeros involves setting the polynomial equal to zero and solving for x. These zeros correspond to the roots or x-intercepts of the polynomial function.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a factor in the polynomial. For example, if (x - a)^k is a factor, then x = a is a zero with multiplicity k. Multiplicity affects the graph's behavior at the zero, such as whether it crosses or touches the x-axis.

Factoring and Solving Quadratic Expressions

Factoring involves expressing a polynomial as a product of simpler polynomials. For quadratic expressions like x^2 - 10, solving for zeros may require techniques such as taking square roots or using the quadratic formula if it cannot be factored easily. This step is essential to find all zeros of the function.