Question

Find the zeroes of the polynomial: $ {x^2} + 2x + 1 $

Answer 1

Hint:
Here we are given the polynomial $ {x^2} + 2x + 1 $ whose zeroes are to be found. Zeroes of the polynomial means that we need to find the value of the variable present in the polynomial at which the polynomial’s value becomes zero. So we need to just equate the above polynomial to zero and get the value or the values of the variable required.

Complete step by step solution:
Here we are given the polynomial $ {x^2} + 2x + 1 $ whose zeroes are to be found. First of all we must be clear with the term polynomial. Polynomial is the expression or the equation that consists of the variables, constants and the exponents joined by the symbol of addition, subtraction and multiplication. For example: $ x + 1 $ is also a polynomial but its degree is one as the highest exponent of the variable in the polynomial is known as the degree of the polynomial.
So we need to find zeros of the polynomial $ {x^2} + 2x + 1 $
So we just need to equate this given polynomial to zero and find the values of the variables at which the polynomial is becoming zero.
This given polynomial $ {x^2} + 2x + 1 $ is the polynomial of the highest exponent $ 2 $ . Hence its degree is $ 2 $
So we will get its two roots but we don’t know about their value. So we need to find them by equating this polynomial to zero.
Equating $ {x^2} + 2x + 1 $ to zero we get:
 $ {x^2} + 2x + 1 = 0 $
We can write $ 2x = x + x $
 $ {x^2} + x + x + 1 = 0 $
Now we can simplify and write
 $ x(x + 1) + 1(x + 1) = 0 $
 $ (x + 1)(x + 1) = 0 $
So we get that $ x + 1 = 0{\text{ and }}x + 1 = 0 $
Which means that both the zeroes of the polynomial are equal
So we get:
 $
  x + 1 = 0 \\
  x = - 1 \\
  $

Hence we get zeroes of the polynomial $ {x^2} + 2x + 1 $ as $ - 1, - 1 $

Note:
If we have the polynomial $ p(x) $ and we get that $ p(m){\text{ and }}p(n) $ are zero, then we can say that $ m{\text{ and }}n $ are the zeroes of the polynomial $ p(x) $ because at these two values the value of the given polynomial is zero.