For each polynomial function, find all zeros and their multiplicities. $\text{ƒ}\left(x\right)=(2x^2-7x+3{)}^3(x-2-\sqrt{5})$

Verified step by step guidance

1. Identify the factors of the polynomial function: \(f(x) = (2x^2 - 7x + 3)^3 (x - 2 - \sqrt{5})\). The zeros come from setting each factor equal to zero.
2. Find the zeros of the quadratic factor \$2x^2 - 7x + 3\( by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \)a=2\(, \)b=-7\(, and \)c=3$.
3. Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots of the quadratic. Then substitute into the quadratic formula to find the two zeros.
4. Note that each zero from the quadratic factor has multiplicity 3 because the entire quadratic is raised to the third power.
5. Find the zero from the linear factor \(x - 2 - \sqrt{5} = 0\), which gives \(x = 2 + \sqrt{5}\), and note that this zero has multiplicity 1 since the factor is to the first power.

Key Concepts

Polynomial Zeros

Zeros of a polynomial are the values of x for which the polynomial equals zero. Finding zeros involves solving the equation f(x) = 0, which may require factoring or using formulas. These zeros represent 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. 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, such as whether it crosses or touches the x-axis.

Factoring and Solving Quadratic Expressions

Factoring quadratic expressions like 2x² - 7x + 3 helps find zeros by rewriting the polynomial as a product of linear factors. When factoring is difficult, the quadratic formula can be used. This step is essential to break down complex polynomials into simpler parts to identify zeros.