Zeroes of a Polynomial
To find the value of a polynomial at x = a, substitute x with a and do the calculation. For example, consider the polynomial eq. To find the value of p(x) at x = 3, replace x with the value 3 and we get
$p\left(3\right)=5\times {3}^{2}2\times 3+3\phantom{\rule{0ex}{0ex}}=456+3\phantom{\rule{0ex}{0ex}}=42$
A real number a is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called the root of the equation p(x) = 0.
Consider the polynomial ${x}^{2}5x+6$. Replace x by 2, then we get
${2}^{2}5\times 2+6=410+6=0$
∴ we say 2 is a zero of the polynomial or 2 is the root of the polynomial ${x}^{2}5x+6$
By checking, we can see that 3 is also a zero of the polynomial.
Note

A zero of a polynomial can be any real number.

0 need not be the zero of a polynomial.

Every linear polynomial has one and only one zero or root.

Every quadratic polynomial has two zeros.

Every cubic polynomial has three zeros.

A polynomial can have one or more zeros.

A nonzero constant polynomial has no zero.
Example 1
Check whether −1 is a zero of the polynomial ${x}^{3}+7x1$
Solution
Replace x by −1.
$({1}^{3})+7\times 11=171$ ≠ 0
∴ −1 is not a zero of the polynomial.
Example 2
Find a zero of the polynomial x +1.
Solution
To find the zero of the polynomial, equate x +1 to 0, i.e., x +1 = 0 ⟹ x = −1.
∴ −1 is a zero of the polynomial
Example 3
Verify whether 2 and 0 are zeroes of the polynomial ${a}^{2}2a$
Solution
$p\left(a\right)={a}^{2}2a\phantom{\rule{0ex}{0ex}}p\left(0\right)={0}^{2}2\times 0=0\phantom{\rule{0ex}{0ex}}p\left(2\right)={2}^{2}2\times 2=0$
Hence, 2 and 0 are the zeroes of the polynomial.