Question

If \[X=\left\{ p:\ where\ p=\dfrac{\left( n+2 \right)\left( 2{{n}^{5}}+3{{n}^{4}}+4{{n}^{3}}+5{{n}^{2}}+6 \right)}{{{n}^{2}}+2n\ and\ n,p\in {{Z}^{+}}} \right\}\] , then find the number of elements?

Answer 1

Hint: In the given question, we have been asked to find the number of elements in the given set X. In order to find the number of elements we need to solve the given polynomial. Then we will need to equate the given polynomial ‘p’ equals to 0. As we know that the polynomial of degree 5 has a total of 5 solutions. So without solving the given equation as it will get very complicated to solve the given polynomial we will just need to find the number of concepts theoretically. In this way we will get the required number of elements in the given set ‘X’.

Complete step-by-step answer:
We have given that,
 \[X=\left\{ p:\ where\ p=\dfrac{\left( n+2 \right)\left( 2{{n}^{5}}+3{{n}^{4}}+4{{n}^{3}}+5{{n}^{2}}+6 \right)}{{{n}^{2}}+2n\ and\ n,p\in {{Z}^{+}}} \right\}\]
We have a polynomial i.e.
 \[p=\dfrac{\left( n+2 \right)\left( 2{{n}^{5}}+3{{n}^{4}}+4{{n}^{3}}+5{{n}^{2}}+6 \right)}{{{n}^{2}}+2n\ and\ n,p\in {{Z}^{+}}}\]
Thus,
 \[\dfrac{\left( n+2 \right)\left( 2{{n}^{5}}+3{{n}^{4}}+4{{n}^{3}}+5{{n}^{2}}+6 \right)}{{{n}^{2}}+2n}=0\]
Taking out the ‘n’ as a common factor from the denominator,
 \[\dfrac{\left( n+2 \right)\left( 2{{n}^{5}}+3{{n}^{4}}+4{{n}^{3}}+5{{n}^{2}}+6 \right)}{n\left( n+2 \right)}=0\]
Cancelling out the common factor i.e. \[\left( n+2 \right)\] from the numerator and denominator,
 \[\dfrac{\left( 2{{n}^{5}}+3{{n}^{4}}+4{{n}^{3}}+5{{n}^{2}}+6 \right)}{n}=0\]
Simplifying the above, we get
 \[\left( 2{{n}^{5}}+3{{n}^{4}}+4{{n}^{3}}+5{{n}^{2}}+6 \right)=0\]
As we know that the polynomial of degree 5 has a total of 5 roots.
So therefore,
The given polynomial has a total of 5 roots.
Here, we do not need to solve the given polynomial because in the given question, we have been asked to find out the number of elements in the set X.
Thus there will be a total of 5 values of ‘p’.
Hence, there will be a total of 5 elements in the given set.
So, the correct answer is “ 5 elements”.

Note: The key point in order to solve the question is that you need to have the basic concepts of polynomials and their roots. Most of the students make mistakes while solving this question as they will try to simplify the equation more and more and it will complicate the answer. Here, only the theoretical concept of a polynomial and the number of roots is required. The polynomial of degree 5 is known as quantic polynomial and it has a characteristic that the polynomial of degree 5 has a total of 5 roots of 5 values. Avoid solving the equation as you will not get the final answer easily.