# Zeroes of a Polynomial

**Consider the polynomial p(x) = 5**

- 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) = 5**

- 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) = 3y3 −4 y+ 11 at y = 2**

- 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) = 4t4 +5 t3−t2+6 at t = a**

- 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.**

**x =**− b a is the only zero of p(x), i.e., a linear polynomial has one and only one zero.

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 p(x) =**x 2 – 2x

**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.**