Factor Theorem
For the polynomial P(x), the factor theorem states that if x =a is zero of P(X) iff x – a is a factor of P(x). i.e., both the following conditions should hold true.
- If a is a zero of P(x) then x−a will be a factor of P(x)
- If x−a is a factor of P(x) then a will be a zero of P(x)
This can be verified by looking at previous examples. Factor theorem can lead to some interesting results, which are as follows:
Result 1: If P(x) is a polynomial of degree “n”, and “r” is a zero of P(x) then P(x) can be written in the following form,
P(x) = (x – r) Q(x)
Where Q(x) is a polynomial of degree “n-1” and can be found out by dividing P(x) with (x – r).
Result 2: If P(x) = (x-r)Q(x) and x = t is a zero of Q(x) then x = t will also be a zero of P(x).
To verify the above fact,
Let’s say “t” is zero Q(x), which means Q(t) = 0.
We know that “r” is a zero of polynomial P(x), where P(x) = (x – r) Q(x),
So we need to check if x = t is also a zero of P(x), let’s put x = t in P(x)
⇒ P(t) = (t – r) Q(t) = 0
So, x = t is also a zero P(x).
Hence, Proved.
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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