# Zero of the Polynomial

The constant polynomial whose coefficients are all equal to 0. The corresponding polynomial function is the constant function with value 0, also called the zero map. The zero polynomial is the additive identity of the additive group of polynomials.

Consider the polynomial p(x) = 5x3– 2x2 + 3x – 2.

If we replace x by 1 everywhere in p(x), we get

p(1) = 5 × (1)3 – 2 × (1)2 + 3 × (1) – 2

= 5 – 2 + 3 –2

= 4

So, we say that the value of p(x) at x = 1 is 4.

Similarly, p(0) = 5(0)3 – 2(0)2 + 3(0) –2 = –2

Example: Find the value of each of the following polynomials at the indicated value of variables:

(i) p(x) = 5x2 – 3x + 7 at x = 1

(ii) q(y) = 3y3 – 4y + √11 at y = 2

(iii) p(t) = 4t4 + 5t3 – t2 + 6 at t = a

Solution:

(i) p(x) =  5x2 – 3x + 7

The value of the polynomial p(x) at x = 1  is given by

p(1) = 5(1)2 – 3(1) + 7 = 5 – 3 + 7 = 9

(ii) q(y) = 3y3 – 4y + √11

The value of the polynomial q(y) at x = 2  is given by

q(2) = 3(2)3 – 4(2) + √11  = 3×8 – 8 + √11   = 24 – 8 + √11  = 16 + √11

(iii) p(t) = 4t4 + 5t3 – t2 + 6 at t = a

The value of the polynomial p(t) at t = a  is given by

p(a) = 4a4 + 5a3 – a2 + 6