For each polynomial function, find all zeros and their multiplicities. ƒ(x)=5x^2(x^2-16)(x+5)

Verified step by step guidance

1. Factor the polynomial completely. Start by recognizing that \(x^2 - 16\) is a difference of squares, which can be factored as \((x - 4)(x + 4)\).
2. Rewrite the function \(f(x) = 5x^2(x^2 - 16)(x + 5)\) as \(f(x) = 5x^2(x - 4)(x + 4)(x + 5)\).
3. Identify the zeros of the polynomial by setting each factor equal to zero: \(5x^2 = 0\), \(x - 4 = 0\), \(x + 4 = 0\), and \(x + 5 = 0\).
4. Solve each equation for \(x\): \(5x^2 = 0\) gives \(x = 0\), \(x - 4 = 0\) gives \(x = 4\), \(x + 4 = 0\) gives \(x = -4\), and \(x + 5 = 0\) gives \(x = -5\).
5. Determine the multiplicity of each zero: \(x = 0\) has multiplicity 2 (from \(5x^2\)), \(x = 4\), \(x = -4\), and \(x = -5\) each have 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 'a' represents coefficients and 'n' is a non-negative integer. 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 also known as roots and can be found by factoring the polynomial or using the quadratic formula. Each zero can have a multiplicity, indicating how many times it is repeated as a solution.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears in the factorization of a 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 of the polynomial and predicting its behavior near the zeros.