For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x+1)^2(x-1)^3(x^2-10)

Verified step by step guidance

1. Identify the factors of the polynomial function: $f(x)=(x+1{)}^2(x-1{)}^3(x^2-10)$.
2. Set each factor equal to zero to find the zeros: $x+1=0$, $x-1=0$, and $x^2-10=0$.
3. Solve each equation: $x+1=0$ gives $x=-1$, $x-1=0$ gives $x=1$, and $x^2-10=0$ gives $x=\pm \sqrt{10}$.
4. Determine the multiplicity of each zero: $x=-1$ has multiplicity 2, $x=1$ has multiplicity 3, and $x=\pm \sqrt{10}$ each have multiplicity 1.
5. Summarize the zeros and their multiplicities: $x=-1$ (multiplicity 2), $x=1$ (multiplicity 3), $x=\sqrt{10}$ (multiplicity 1), $x=-\sqrt{10}$ (multiplicity 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 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 critical as they indicate where the graph of the polynomial intersects the x-axis. Finding zeros often involves factoring the polynomial or using the quadratic formula for quadratic factors.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. If a zero has an even multiplicity, the graph touches the x-axis at that point, while an odd multiplicity means the graph crosses the x-axis. Understanding multiplicity helps in sketching the graph and predicting its behavior near the zeros.