If, X={ p: where p=((n+2)(2 n^5+3 n^4+4 n^3+5 n^2+6))/(n^2+2 n) and n, p ∈ Z^+ }, th

step 1 Hello students and this problem we are given a set x which is defined on p where p is equals to

If, X={ p: where p=((n+2)(2 n^5+3 n^4+4 n^3+5...

If, $X=\left\{\begin{array}{c}p: \text { where } p=\frac{(n+2)\left(2 n^{5}+3 n^{4}+4 n^{3}+5 n^{2}+6\right)}{n^{2}+2 n} \\ \text { and } n, p \in Z^{+}\end{array}\right\}$, then find the number of elements
in the set $\mathrm{X}$.
(1) 2
(2) 3
(3) 4
(4) 6

Key Concepts

Divisibility in Rational Expressions

This concept involves analyzing when a ratio of two polynomial expressions yields an integer value. By ensuring that the numerator is divisible by the denominator, one can determine conditions on the variables that permit the fraction to simplify to an integer.

Polynomial Factorization and Simplification

Factorization is used to simplify complex algebraic expressions, making them easier to analyze. Simplifying by canceling common factors often reveals underlying divisibility properties and simplifies the task of determining integer solutions.

Divisor Constraints in Diophantine Equations

Diophantine problems require solutions in integers, which means that divisibility conditions play a key role. By identifying factors that must divide a given term, one can restrict the set of possible integer values, reducing an infinite search space to a few candidates.

Counting Elements in Finite Sets

After determining the necessary constraints for integer solutions, the next step is to count the number of distinct solutions that meet those conditions. This involves ensuring that no extra or duplicate values are included, thereby accurately determining the set's cardinality.

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