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  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) = u2 – 5u + 6;
 g(3) = (3)2 – 5 × 3 + 6 = 9 – 15 + 6 = 0
∴ 3 is a zero of the polynomial g(u) = u2 – 5u + 6.
Also, g(2) = (2)2 – 5 × 2 + 6 = 4 – 10 + 6 = 0
∴ 2 is also a zero of the polynomial g(u) = u2 – 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) = x2 – 4, then f(2) = (2)2 – 4 = 4 – 4 = 0
    Here, zero of the polynomial f(x) = x2 – 4 is 2 which itself is not 0.
  • 0 may be a zero of a polynomial.
    F or example : If f(x) = x2 – x, 
    then f(0) = 02 – 0 = 0
    Here 0 is the zero of polynomial
    f(x) = x2 – x.

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