What is the multiple zero and multiplicity of f(x) = x3 + 2x2 + x?

Number System is a method of representing numbers on a number line. The symbols range from 0-9 and are termed digits. A polynomial is a function of the form f(x) = aⁿ xⁿ + aⁿ⁻¹ xⁿ⁻¹ + … + a²x² + a¹x + a⁰. The degree of a polynomial is the highest power of x in the expression. Constant (non-zero) polynomials are of degree 0, linear polynomials (maximum power of x is 1) are of degree 1, Quadratic polynomials (maximum power of x is 2) are of degree 2 and so on. 

Roots or Zeroes

If a and b are roots, then the polynomial function with these roots is f(x) = (x – a)(x – b), or a multiple of this. For example, if a quadratic expression has the roots x = 3 and x = -2, then the function must be f(x) = (x – 3)(x + 2), or a constant multiple of this. This can be applied to polynomials of any degree. For example, if the roots of a polynomial are x = 2, x = 3, x = 4, then the function must be f(x) = (x – 2)(x – 3)(x – 4), or a constant multiple of this. Lets also try to think about the function f(x) = (x – 1)² . It is seen that x – 1 = 0, so x = 1. For this function, there is a root. This is what is called a repeated root and this root can be repeated any number of times. For example, f(x) = (x – 2)³(x + 4)⁴ has a repeated root x = 2, and a repeated root x = -4. It can be said that the root x = 2 has a multiplicity of 3 and that the root x = -4 has a multiplicity of 4. 

Multiplicity and Multiple Roots

Multiple roots of a polynomial are roots whose factors show up more than once in the complete factorization of the polynomial. We call the number of times a factor shows up in the complete factorization the multiplicity of the root. The following examples will demonstrate how multiplicity and multiple roots are found.

What is the multiple zero and multiplicity of f(x) = x³ +  2x²  + x?

Solution:

The Objective here to is to find the Multiple zero and multiplicity of f(x) = x³ + 2x² + x. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. A multiple zero is a root with multiplicity m ≥ 2.

f(x) = x³ + 2x² + x. Will be equated to zero. 

x³ + 2x² + x = 0

x(x² + 2x + 1) = 0 (extract x common from the equation and the remaining part becomes a quadratic equation)

x² + 2x + 1 can be written as (x + 1)² it can be seen that the roots or zeroes of f(x) are 0, -1. Here zero has a multiplicity of 1 since it occurs once in the factored form. -1 has a multiplicity of 2. Therefore, multiple zero of f(x) = x³ + 2x² + x, is -1 and it has multiplicity of 2.

Similar Problems

Question 1: What is the multiple zero and multiplicity of y = 3(x + 3)³ (x + 2)⁴ (x – 1)² (x – 5).

Solution:

Roots of this function are,

x + 3 = 0 -> x = -3

x + 2 = 0 -> x = -2

x – 1 = 0 -> x = 1

x – 5 = 0 -> x = 5

Multiple zeroes are -5, -2, 1. Multiplicity of x = -5 is 3 because x + 5 is raised to the power 3, Similarly, x = -2 is 4 and x = 1 is 2.

Question 2: What is the multiple zero and multiplicity of y = (x + 1)²(x + 3)³

Solution:

Roots of this function are,

x + 1 = 0 -> x = -1

x + 3 = 0 -> x = -3F

Multiple zeroes are -1, 3. The multiplicity of x = -1 is 2 because x + 1 term is raised to the power 2 and multiplicity of x = -3 is 3 because x + 3 is raised to the power 3