For each polynomial function, find all zeros and their multiplicities. $\text{ƒ}\left(x\right)=(x^2+x-2{)}^5(x-1+\sqrt{3}{)}^2$

Verified step by step guidance

1. Start by identifying the factors of the polynomial function: \(f(x) = (x^2 + x - 2)^5 (x - 1 + \sqrt{3})^2\).
2. Find the zeros of the first factor \(x^2 + x - 2\) by solving the quadratic equation \(x^2 + x - 2 = 0\). Use factoring, completing the square, or the quadratic formula.
3. Once you find the roots of \(x^2 + x - 2 = 0\), note that each root has a multiplicity of 5 because the entire quadratic factor is raised to the 5th power.
4. Next, find the zero of the second factor \(x - 1 + \sqrt{3} = 0\) by isolating \(x\), which gives \(x = 1 - \sqrt{3}\). This zero has a multiplicity of 2 since the factor is squared.
5. List all zeros found along with their multiplicities: the roots from the quadratic factor each with multiplicity 5, and the root from the linear factor with multiplicity 2.

Key Concepts

Polynomial Zeros

Zeros of a polynomial are the values of x that make the polynomial equal to zero. Finding zeros involves solving the equation f(x) = 0, which can be done by factoring or using other algebraic methods. Each zero corresponds to a root of the polynomial.

Multiplicity of Zeros

Multiplicity refers to how many times a particular zero appears as a factor in the polynomial. If a factor is raised to a power n, the zero associated with that factor has multiplicity n. Multiplicity affects the graph's behavior at the zero.

Factoring and Solving Quadratic Expressions

Factoring quadratic expressions like x^2 + x - 2 helps find zeros by rewriting the polynomial as a product of linear factors. Solving these factors set to zero gives the roots. Recognizing and factoring quadratics is essential for breaking down complex polynomials.