Zeroes of a Polynomial
Consider the polynomial p(x) =
- If we replace x by 1 everywhere in p(x), we get
- calculate the terms
- Therfore, p(1) =
- Also find the p(0) =
- calculate the terms
- Therfore, p(0) =
Example 2
Find the value of each of the following polynomials at the indicated value of variables:
(i) p(x) =
- The value of the polynomial p(x) at x = 1 is given by
- calculate the terms
- Therfore, p(x) =
- We have found the answer.
(ii) q(y) =
- The value of the polynomial q(y) at y = 2 is given by
- calculate the terms
- Therfore, q(y) =
+ - We have found the answer.
(iii) p(t) =
- The value of the polynomial p(t) at t = a is given by
- Calculating the terms, we get: p(a) =
+ +a 3 + 6. - We have found the answer.
Now, consider the polynomial p(x) = x – 1.
What is p(1)? Note that : p(1) = 1 – 1 =
As p(1) = 0, we say that 1 is a zero of the polynomial p(x).
Similarly, you can check that 2 is a zero of q(x), where q(x) = x – 2.
In general, we say that a zero of a polynomial p(x) is a number c such that p (c) = 0. In other words,
Zeros of a Polynomial are those values when put in the polynomial instead of a variable, the result becomes zero.
You must have observed that the zero of the polynomial x – 1 is obtained by equating it to 0, i.e., x – 1 = 0, which gives x =
We say p(x) = 0 is a polynomial equation and 1 is the root of the polynomial equation p(x) = 0. So we say 1 is the zero of the polynomial x – 1, or a root of the polynomial equation x – 1 = 0.
Now, consider the constant polynomial 5. Can you tell what its zero is?
It has no zero of polynomial because replacing x by any number in
In fact, a non-zero constant polynomial has no zero. What about the zeroes of the zero polynomial? By convention, every real number is a zero of the zero polynomial.
Note: Do not get confused between
Example 3
Check whether –2 and 2 are zeroes of the polynomial x + 2
Example 4
Find a zero of the polynomial p(x) = 2x + 1.
Now: We can say that 1 is the zero of x – 1, and –2 is the zero of x + 2.
Example 5
Verify whether 2 and 0 are zeroes of the polynomial
Let us now list our observations:
(i) A zero of a polynomial need not be 0.
(ii) 0 may be a zero of a polynomial.
(iii) Every linear polynomial has one and only one zero.
(iv) A polynomial can have more than one zero.