Fundamental Theorem of Linear Algebra
If P(x) is a polynomial of degree “n” then P(x) will have exactly n zeros, some of which may repeat.
This means that if we list out all the zeroes and list each one k times when k is its multiplicity. We will have exactly n numbers in the list. This can be useful as it can give us an idea about how many zeros should be there in a polynomial. So we can stop looking for zeros once we reach our required number of zeros.
Multiplicity of a Root
Suppose we have a polynomial P(x) = 0 which factorizes into,
P(x) = (x – r)k(x – a)m
If r is a zero of a polynomial and the exponent on its term that produced the root is k then we say that r has multiplicity k. Zeroes with a multiplicity of 1 are often called simple zeroes and zeros with a multiplicity of 2 are called double roots of the polynomial.
Example: P(x) is a degree-5 polynomial, that has been factorized for you. List the roots and their multiplicity.
P(x) = 5x5−20x4+5x3+50x2−20x−40=5(x+1)2(x−2)3
Solution:
Given, P(x) = 5(x+1)2(x−2)3
⇒ P(x) = 5(x+1)(x+1)(x+1)(x−2)(x−2)
To find zeros, P(x) = 0
⇒ x = -1, -1, 2, 2, 2
Notice that “-1” occurs two times as a zero, so its multiplicity is 2 while the multiplicity of the zero “2” is 3.
Articles related to Zeros of Polynomial
Zeros of Polynomial
Zeros of a Polynomial are those real, imaginary, or complex values when put in the polynomial instead of a variable, the result becomes zero (as the name suggests zero as well). Polynomials are used to model some physical phenomena happening in real life, they are very useful in describing situations mathematically.
The zeros of a polynomial are all the x-values that make the polynomial equal to zero. Zeroes of a polynomial tell us about the x-intercepts of the polynomial’s graph. In this article, we will discuss about the zeroes of a polynomial, how to find them, the factor theorem, etc.
Table of Content
- What are Zeros of Polynomials?
- Zeros of Polynomial Formula
- How to Find Zero of a Polynomial?
- For Linear Polynomial
- For Quadratic Polynomial
- For Cubic Polynomial
- Factor Theorem
- Relation between Zeros and Coefficient
- Relation between Zeros and Coefficient for Quadratic Equation
- Relation between Zeros and Coefficient for Cubic Equation
- Forming Equation with Zeros of Polynomial
- Zeros in Graph of Polynomials
- Fundamental Theorem of Linear Algebra
- Multiplicity of a Root
- Articles related to Zeros of Polynomial
- Sample Problems on Zeros of Polynomial
- Practice Problems on Zeros of Polynomial
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