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.