Question

The remainder when \[{x^2} + 2x + 1\] is divided by \[\left( {x + 1} \right)\] is
A) 4
B) 0
C) 1
D) \[ - 2\]

Answer 1

Hint:
Here, we will use the remainder theorem, which tells us that when we divide a polynomial \[f\left( x \right)\] by \[x - c\], the remainder is \[f\left( c \right)\]. We will rewrite the given divisor and then compare the equation \[x - c\] and the obtained divisor to find the value of \[c\]. Then substitute the value of \[c\] in the given polynomial \[f\left( x \right)\] to find the required value.

Complete step by step solution:
We are given that the polynomial is \[{x^2} + 2x + 1\] is divided by \[\left( {x + 1} \right)\].
Let us assume that \[f\left( x \right) = {x^2} + 2x + 1\].
We know that in the remainder theorem, when we divide a polynomial \[f\left( x \right)\] by \[x - c\], the remainder is \[f\left( c \right)\].
Rewriting the given divisor \[\left( {x + 1} \right)\], we get
\[ \Rightarrow x - \left( { - 1} \right)\]
We will first find the value of \[c\] by comparing the equation \[x - c\] and the above divisor, we get
\[ \Rightarrow c = - 1\]
Substituting the value of \[c\] in the given polynomial \[f\left( x \right)\], we get
\[
   \Rightarrow f\left( { - 1} \right) = {\left( { - 1} \right)^2} + 2\left( { - 1} \right) + 1 \\
   \Rightarrow f\left( { - 1} \right) = 1 - 2 + 1 \\
   \Rightarrow f\left( { - 1} \right) = 0 \\
 \]
Thus, the remainder is 0.

Hence, option B is correct.

Note:
In solving these types of questions, students should be careful while calculations. We can also find the remainder by using the division algorithm method, by dividing the first term of the quotient with the highest term of the dividend, which is really time-consuming and is used when asked in the question. Since in the question we are asked to find the remainder, so students are advised to solve this question using the remainder theorem.
Dividing the \[{x^2} + 2x + 1\] by \[\left( {x + 1} \right)\] using the long division method, we get

Thus, the remainder is 0.