Zeros Multiplicity and Degrees Calculator

What Are Zeros, Multiplicity, and Degrees?

In algebra, a polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. The degree of a polynomial is the highest power of the variable in the polynomial.

A zero (or root) of a polynomial is a value of the variable that makes the polynomial equal to zero. The multiplicity of a zero is the number of times the zero occurs as a root. For example, if a polynomial has a root at x = 2 and it touches the x-axis at that point, the multiplicity is 2.

Key Terms

• Zero (Root): A solution to the equation P(x) = 0.
• Multiplicity: The number of times a zero occurs.
• Degree: The highest power of the variable in the polynomial.

How to Calculate Zeros, Multiplicity, and Degrees

To calculate the zeros, their multiplicities, and the degree of a polynomial, follow these steps:

1. Identify the polynomial equation.
2. Factor the polynomial to express it in terms of its roots.
3. Count the multiplicity of each root by determining the highest power of each factor.
4. Identify the highest degree term in the polynomial.

For example, consider the polynomial P(x) = (x - 2)³(x + 1)². Here, the zeros are x = 2 and x = -1. The multiplicity of x = 2 is 3, and the multiplicity of x = -1 is 2. The degree of the polynomial is 5 (since 3 + 2 = 5).

Example Calculation

Let's analyze the polynomial P(x) = (x - 3)²(x + 4).

1. Identify the zeros: x = 3 and x = -4.
2. Determine the multiplicity of each zero:
   • x = 3 has multiplicity 2 (since (x - 3)² is a factor).
   • x = -4 has multiplicity 1 (since (x + 4) is a factor).
3. Calculate the degree: 2 (from the first factor) + 1 (from the second factor) = 3.

Thus, the polynomial has zeros at x = 3 (multiplicity 2) and x = -4 (multiplicity 1), and its degree is 3.