Theory:

If the value of the polynomial \(p(x)\) at \(x = k\) is zero \(p(k) = 0\), then the real number \(k\) is called the zero of the polynomial \(p(x)\).
Important!
To find the zeroes of a quadratic polynomial, put \(p(x) = 0\).
Example:
1. Find the zeroes of the quadratic expression \(p(x) = 2x^2 + 3x + 1\).
 
Solution:
 
\(p(x) = 2x^2 + 3x + 1\)
 
Put \(p(x) = 0\).
 
\(2x^2 + 3x + 1 = 0\)
 
\(2x^2 + 2x + x + 1 = 0\)
 
\(2x(x + 1) + 1(x + 1) = 0\)
 
\((2x + 1) (x + 1) = 0\)
 
\(2x + 1 = 0\) or \(x + 1 = 0\)
 
\(x = \frac{-1}{2}\) or \(x = -1\)
 
Thus, the zeroes of \(p(x) = 2x^2 + 3x + 1\) are \(\frac{-1}{2}\) and \(-1\).
 
 
2. Check \(2\) is a zero of the polynomial \(p(x) = 3x^2 - 5x - 2\).
 
Solution:
 
\(p(x) = 3x^2 - 5x - 2\)
 
Put \(x = 2\) in \(p(x)\).
 
\(p(2) = 3(2)^2 - 5(2) - 2\)
 
\(= 3(4) - 10 - 2\)
 
\(= 12 - 12\)
 
\(= 0 \)
 
Thus, \(2\) is a zero of the polynomial \(p(x) = 3x^2 - 5x - 2\).