Theory:

The value of the polynomial \(p(x)\) at \(x=a\) is \(p(a)\) acquired when \(x\) is replaced by \(a\) (\(a∈R\)).
Example:
Write the value of \(p(x) = x^2+2x-1\) at \(x = 3\).
 
The value of \(p(x)\) at \(x = 3\) can be obtained by substituting the point \(x = 3\) in the polynomial.
 
Substitute \(x=3\) in the polynomial \(p(x)\).
 
\(p(3) = \)\(3^2+2(3)-1\)
 
\(=9+6-1\)
 
\(=14\).
Zero of polynomial \(p(x)\) is a real number '\(a\)' such that \(p(a)= 0\).
For polynomial \(p(x)\), if \(p(a) = 0\) then the zero polynomial is \(x = a\).
 
So, we put \(p(x) = 0\) to find zero polynomial.
 
The value of \(x\) is then found.
 
Important!
1. Zero of a polynomial: To find the zero of polynomial, we put \(p(x) =\) \(0\) in the given polynomial \(p(x)\).
    Example:
    i. Consider \(p(x)=3a\)
     
    Putting \(p(x) = 0\) in the polynomial equation.
     
    0 = 3a.0/3 =a0=a
     
    Thus, \(a=0\) is the zero of the polynomial \(p(a)=3a\).
     
     
    ii. Consider the polynomial p(x)=x2.
     
    Putting \(p(x) = 0\) in the polynomial equation.
     
    0=x2x=2
     
    Thus \(x=2\) is the zero of the polynomial p(x)=x2.
2. Zeros of a polynomial p(x) is the real number 'a' for which p(x) if p(a)=0. In this situation, we say \(p(x) = 0\) is a polynomial equation, and a is a root of the polynomial.
  • Each real number is a zero of the zero polynomial p(x)=0.
Example:
         The polynomial \(p(8) = 0\), where \(8\) is a real number.
  • The non -zero constant polynomial does not have zeros.
Example:
          The polynomial p(x)=8 does not have a Zero.