Zeros Of A Polynomial Function
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
(a) Every linear polynomial has one and only one zero.
(b) A given polynomial may have more than one zeroes.
(c) 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.
(d) 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.
(e) 0 may be a zero of a polynomial.
For Example: If f(x) = x2 – x,
then f(0) = 02 – 0 = 0
Here 0 is the zero of polynomial
f(x) = x2 – x.
Zeros Of A Polynomial Function With Examples
Example 1: Verify whether the indicated numbers are zeroes of the polynomial corresponding to them in the following cases :
(i) p(x) = 3x + 1, x = \(-\frac{1}{3}\)
(ii) p(x) = (x + 1) (x – 2), x = – 1, 2
(iii) p(x) = x2, x = 0
(iv) p(x) = lx + m, x = \(-\frac{m}{l}\)
(v) p(x) = 2x + 1, x = \(\frac{1}{2}\)
Sol.
(i) p(x) = 3x + 1
\(\Rightarrow p\left( -\frac{1}{3} \right)=3\times -\frac{1}{3}+1=-1+1=0\)
∴ x = \(-\frac{1}{3}\) is a zero of p(x) = 3x + 1.
(ii) p(x) = (x + 1) (x – 2)
⇒ p(–1) = (–1 + 1) (–1 – 2) = 0 × –3 = 0
and, p(2) = (2 + 1) (2 – 2) = 3 × 0 = 0
∴ x = –1 and x = 2 are zeroes of the given polynomial.
(iii) p(x) = x2
⇒ p(0) = 02 = 0
∴ x = 0 is a zero of the given polynomial
(iv) p(x) = lx + m
\(\Rightarrow p\left( -\frac{m}{l} \right)=l\left( -\frac{m}{l} \right)+m\)
= – m + m = 0
∴ x = \(-\frac{m}{l}\) is a zero of the given polynomial.
(v) p(x) = 2x + 1
\(\Rightarrow p\left( \frac{1}{2} \right)=2\times \frac{1}{2}+1\)
= 1 + 1 = 2 ≠ 0
∴ x = \(\frac{1}{2}\) is not a zero of the given polynomial.
Example 2: Find the zero of the polynomial in each of the following cases :
(i) p(x) = x + 5
(ii) p(x) = 2x + 5
(iii) p(x) = 3x – 2
Sol.
To find the zero of a polynomial p(x) means to solve the polynomial equation p(x) = 0.
(i) For the zero of polynomial p(x) = x + 5
p(x) = 0 ⇒ x + 5 = 0 ⇒ x = –5
∴ x = –5 is a zero of the polynomial.
p(x) = x + 5.
(ii) p(x) = 0 ⇒ 2x + 5 = 0
⇒ 2x = –5 and x = \(-\frac{5}{2}\)
∴ \(-\frac{5}{2}\) is a zero of p(x) = 2x + 5.
(iii) p(x) = 0 ⇒ 3x – 2 = 0
⇒ 3x = 2 and x = \(\frac{2}{3}\).
∴ x = \(\frac{2}{3}\) is zero of p(x) = 3x – 2.