#### Mathematics (NTSE/Olympiad)

# Polynomials

__Zeros of a Polynomial__

**If for x = a, the value of the polynomial p(x) is 0 i.e., p(a) = 0; then x = a is a zero of the polynomial p(x).**

For example :

(i) For polynomial p(x) = x – 2;

p(2) = 2 – 2 = 0

∴ x = 2 or simply 2 is a zero of the polynomial p(x) = x – 2.

(ii) For the polynomial g(u) = u^{2} – 5u + 6;

g(3) = (3)^{2} – 5 × 3 + 6 = 9 – 15 + 6 = 0

∴ 3 is a zero of the polynomial g(u) = u^{2} – 5u + 6.

Also, g(2) = (2)^{2} – 5 × 2 + 6 = 4 – 10 + 6 = 0

∴ 2 is also a zero of the polynomial g(u) = u^{2} – 5u + 6

**Impoortant Points to Remember: **

- Every linear polynomial has one and only one zero.
- A given polynomial may have more than one zeroes.
- If the degree of a polynomial is n; the largest number of zeroes it can have is also n.

**For example :**

If the degree of a polynomial is 5, the polynomial can have at the most 5 zeroes; if the degree of a polynomial is 8; largest number of zeroes it can have is 8. - A zero of a polynomial need not be 0.

**For example : If f(x) = x**^{2} – 4, then f(2) = (2)^{2} – 4 = 4 – 4 = 0

Here, zero of the polynomial f(x) = x^{2} – 4 is 2 which itself is not 0. - 0 may be a zero of a polynomial.

**F or example : If f(x) = x**^{2} – x,

then f(0) = 0^{2} – 0 = 0

Here 0 is the zero of polynomial

f(x) = x^{2} – x.