Question

Check whether $ \left( {x + 1} \right) $ is a factor of $ {x^4} + {x^3} + {x^2} + x + 1 $ .
(A) True
(B) False

Answer 1

Hint: In the given question, we are required to find whether the given function in x $ {x^4} + {x^3} + {x^2} + x + 1 $ is divisible by the divisor $ \left( {x + 1} \right) $ or not. The question can be solved with the help of remainder theorem. This question requires us to have the knowledge of basic and simple algebraic rules and operations such as substitution, addition, multiplication, subtraction and many more like these. A thorough understanding of functions, division algorithms and its applications will be of great significance.

Complete step-by-step answer:
So, the dividend function is $ {x^4} + {x^3} + {x^2} + x + 1 $ .
Divisor function is $ \left( {x + 1} \right) $ ,
Now, we have to check whether the divisor function $ \left( {x + 1} \right) $ is a factor of the dividend function $ {x^4} + {x^3} + {x^2} + x + 1 $ .
We can check this with the help of remainder theorem. Remainder theorem helps us in calculating the remainder when the dividend function is divided by the divisor function by putting in a certain value of variable in the dividend function. So, if we get the remainder as zero, then we can conclude that the function $ \left( {x + 1} \right) $ is a factor of $ {x^4} + {x^3} + {x^2} + x + 1 $ .
First equating the divisor $ \left( {x + 1} \right) $ as zero, we get the value of x as,
 $ \Rightarrow \left( {x + 1} \right) = 0 $
 $ \Rightarrow x = - 1 $
So, we substitute the value of x as $ \left( { - 1} \right) $ in the original dividend function so as to get the remainder using the remainder theorem.
Hence, we get the remainder as
 $ \Rightarrow {\left( { - 1} \right)^4} + {\left( { - 1} \right)^3} + {\left( { - 1} \right)^2} + \left( { - 1} \right) + 1 $
Now, evaluating the powers of $ \left( { - 1} \right) $ , we get,
 $ \Rightarrow 1 + \left( { - 1} \right) + 1 + \left( { - 1} \right) + 1 $
Opening the brackets, we get,
\[ \Rightarrow 1 - 1 + 1 - 1 + 1\]
Cancelling the like terms with opposite signs, we get,
\[ \Rightarrow 1\]
Hence, we get the remainder obtained on dividing $ {x^4} + {x^3} + {x^2} + x + 1 $ by the divisor $ \left( {x + 1} \right) $ as $ 1 $ .
Therefore, $ \left( {x + 1} \right) $ is not a factor of $ {x^4} + {x^3} + {x^2} + x + 1 $ .
So, option (B) is correct.
So, the correct answer is “Option B”.

Note: Remainder theorem is an approach of Euclidean division of polynomials. Remainder theorem requires just a simple change of variable in the function so as to find the remainder of the division procedure. Substitution of a variable involves putting a certain value in place of the variable. That specified value may be a certain number or even any other variable.