# How to Solve this specific inequality

I want to solve this inequality for l. I have tried to use Reduce and Solve and its not worked out. How can I solve this?

Solve[((n*p - l - 1)*(n*q - l -
1)*(n*p - l)^3*(n*q - l)^3) < (l*(n*(1 - q) - n*p +
l)*(l + 1)^3*(n*(1 - q) - n*p + l + 1)^3), l]


Let me write this inequality normally as well $$(np-l-1)(nq-l-1)(np-l)^3(nq-l)^3

• The command Collect[Expand[((n*p - l - 1)*(n*q - l - 1)*(n*p - l)^3*(n*q - l)^3) < (l*(n*(1 - q) - n*p + l)*(l + 1)^3*(n*(1 - q) - n*p + l + 1)^3)], l] shows you deal with a polynomial inequality of degree 7 depending on three parameters n,p,q (I think that p+q==1,p>=0,p<=1,n>0.). You can numerically solve the equation, evaluating the parameters, and then determine the intervals where the inequality holds. Jun 18, 2020 at 8:36

Since both side of the inequality can be set zero independently and both are strictly factored the zero are evident. The problem for visualization is there are no numerical values to solve.

Another perspective is that there is only one inequality for {l,n,p,q} for unknowns and that is a situation Solve and Root for example are not really good at. Solve needs are vast of time. Roots is faster, but the solution is already not nice to reads and does not make use of the beautiful factorization.

Roots[((n*p - l - 1)*(n*q - l -
1)*(n*p - l)^3*(n*q - l)^3) - (l*(n*(1 - q) - n*p +
l)*(l + 1)^3*(n*(1 - q) - n*p + l + 1)^3) == 0, l,
Method -> "JenkinsTraub"]

l == Root[-n^6 p^3 q^3 + n^7 p^4 q^3 + n^7 p^3 q^4 -
n^8 p^4 q^4 + (n + 3 n^2 + 3 n^3 + n^4 - n p - 6 n^2 p -
9 n^3 p - 4 n^4 p + 3 n^2 p^2 + 9 n^3 p^2 + 6 n^4 p^2 -
3 n^3 p^3 - 4 n^4 p^3 + n^4 p^4 - n q - 6 n^2 q - 9 n^3 q -
4 n^4 q + 6 n^2 p q + 18 n^3 p q + 12 n^4 p q - 9 n^3 p^2 q -
12 n^4 p^2 q + 4 n^4 p^3 q + 3 n^2 q^2 + 9 n^3 q^2 +
6 n^4 q^2 - 9 n^3 p q^2 - 12 n^4 p q^2 + 6 n^4 p^2 q^2 +
3 n^5 p^3 q^2 - 3 n^6 p^4 q^2 - 3 n^3 q^3 - 4 n^4 q^3 +
4 n^4 p q^3 + 3 n^5 p^2 q^3 - 8 n^6 p^3 q^3 + 4 n^7 p^4 q^3 +
n^4 q^4 - 3 n^6 p^2 q^4 + 4 n^7 p^3 q^4) #1 + (1 + 9 n +
18 n^2 + 13 n^3 + 3 n^4 - 9 n p - 36 n^2 p - 39 n^3 p -
12 n^4 p + 18 n^2 p^2 + 39 n^3 p^2 + 18 n^4 p^2 -
13 n^3 p^3 - 12 n^4 p^3 + 3 n^4 p^4 - 9 n q - 36 n^2 q -
39 n^3 q - 12 n^4 q + 36 n^2 p q + 78 n^3 p q + 36 n^4 p q -
39 n^3 p^2 q - 36 n^4 p^2 q + 9 n^4 p^3 q + 3 n^5 p^4 q +
18 n^2 q^2 + 39 n^3 q^2 + 18 n^4 q^2 - 39 n^3 p q^2 -
36 n^4 p q^2 + 9 n^4 p^2 q^2 + 18 n^5 p^3 q^2 -
6 n^6 p^4 q^2 - 13 n^3 q^3 - 12 n^4 q^3 + 9 n^4 p q^3 +
18 n^5 p^2 q^3 - 16 n^6 p^3 q^3 + 3 n^4 q^4 + 3 n^5 p q^4 -
6 n^6 p^2 q^4) #1^2 + (6 + 30 n + 42 n^2 + 21 n^3 + 3 n^4 -
30 n p - 84 n^2 p - 63 n^3 p - 12 n^4 p + 42 n^2 p^2 +
63 n^3 p^2 + 18 n^4 p^2 - 20 n^3 p^3 - 12 n^4 p^3 +
2 n^4 p^4 - 30 n q - 84 n^2 q - 63 n^3 q - 12 n^4 q +
84 n^2 p q + 126 n^3 p q + 36 n^4 p q - 54 n^3 p^2 q -
36 n^4 p^2 q - 4 n^4 p^3 q + 4 n^5 p^4 q + 42 n^2 q^2 +
63 n^3 q^2 + 18 n^4 q^2 - 54 n^3 p q^2 - 36 n^4 p q^2 -
18 n^4 p^2 q^2 + 24 n^5 p^3 q^2 - 20 n^3 q^3 - 12 n^4 q^3 -
4 n^4 p q^3 + 24 n^5 p^2 q^3 + 2 n^4 q^4 +
4 n^5 p q^4) #1^3 + (15 + 50 n + 48 n^2 + 15 n^3 + n^4 -
50 n p - 96 n^2 p - 45 n^3 p - 4 n^4 p + 45 n^2 p^2 +
45 n^3 p^2 + 6 n^4 p^2 - 10 n^3 p^3 - 4 n^4 p^3 - 50 n q -
96 n^2 q - 45 n^3 q - 4 n^4 q + 87 n^2 p q + 90 n^3 p q +
12 n^4 p q - 15 n^3 p^2 q - 12 n^4 p^2 q - 12 n^4 p^3 q +
45 n^2 q^2 + 45 n^3 q^2 + 6 n^4 q^2 - 15 n^3 p q^2 -
12 n^4 p q^2 - 30 n^4 p^2 q^2 - 10 n^3 q^3 - 4 n^4 q^3 -
12 n^4 p q^3) #1^4 + (20 + 45 n + 27 n^2 + 4 n^3 - 42 n p -
54 n^2 p - 12 n^3 p + 18 n^2 p^2 + 12 n^3 p^2 - 42 n q -
54 n^2 q - 12 n^3 q + 30 n^2 p q + 24 n^3 p q +
12 n^3 p^2 q + 18 n^2 q^2 + 12 n^3 q^2 +
12 n^3 p q^2) #1^5 + (14 + 21 n + 6 n^2 - 14 n p - 12 n^2 p -
14 n q - 12 n^2 q - 4 n^2 p q) #1^6 + (4 + 4 n) #1^7 &, 1] ||
l == Root[-n^6 p^3 q^3 + n^7 p^4 q^3 + n^7 p^3 q^4 -
n^8 p^4 q^4 + (n + 3 n^2 + 3 n^3 + n^4 - n p - 6 n^2 p -
9 n^3 p - 4 n^4 p + 3 n^2 p^2 + 9 n^3 p^2 + 6 n^4 p^2 -
3 n^3 p^3 - 4 n^4 p^3 + n^4 p^4 - n q - 6 n^2 q - 9 n^3 q -
4 n^4 q + 6 n^2 p q + 18 n^3 p q + 12 n^4 p q - 9 n^3 p^2 q -
12 n^4 p^2 q + 4 n^4 p^3 q + 3 n^2 q^2 + 9 n^3 q^2 +
6 n^4 q^2 - 9 n^3 p q^2 - 12 n^4 p q^2 + 6 n^4 p^2 q^2 +
3 n^5 p^3 q^2 - 3 n^6 p^4 q^2 - 3 n^3 q^3 - 4 n^4 q^3 +
4 n^4 p q^3 + 3 n^5 p^2 q^3 - 8 n^6 p^3 q^3 + 4 n^7 p^4 q^3 +
n^4 q^4 - 3 n^6 p^2 q^4 + 4 n^7 p^3 q^4) #1 + (1 + 9 n +
18 n^2 + 13 n^3 + 3 n^4 - 9 n p - 36 n^2 p - 39 n^3 p -
12 n^4 p + 18 n^2 p^2 + 39 n^3 p^2 + 18 n^4 p^2 -
13 n^3 p^3 - 12 n^4 p^3 + 3 n^4 p^4 - 9 n q - 36 n^2 q -
39 n^3 q - 12 n^4 q + 36 n^2 p q + 78 n^3 p q + 36 n^4 p q -
39 n^3 p^2 q - 36 n^4 p^2 q + 9 n^4 p^3 q + 3 n^5 p^4 q +
18 n^2 q^2 + 39 n^3 q^2 + 18 n^4 q^2 - 39 n^3 p q^2 -
36 n^4 p q^2 + 9 n^4 p^2 q^2 + 18 n^5 p^3 q^2 -
6 n^6 p^4 q^2 - 13 n^3 q^3 - 12 n^4 q^3 + 9 n^4 p q^3 +
18 n^5 p^2 q^3 - 16 n^6 p^3 q^3 + 3 n^4 q^4 + 3 n^5 p q^4 -
6 n^6 p^2 q^4) #1^2 + (6 + 30 n + 42 n^2 + 21 n^3 + 3 n^4 -
30 n p - 84 n^2 p - 63 n^3 p - 12 n^4 p + 42 n^2 p^2 +
63 n^3 p^2 + 18 n^4 p^2 - 20 n^3 p^3 - 12 n^4 p^3 +
2 n^4 p^4 - 30 n q - 84 n^2 q - 63 n^3 q - 12 n^4 q +
84 n^2 p q + 126 n^3 p q + 36 n^4 p q - 54 n^3 p^2 q -
36 n^4 p^2 q - 4 n^4 p^3 q + 4 n^5 p^4 q + 42 n^2 q^2 +
63 n^3 q^2 + 18 n^4 q^2 - 54 n^3 p q^2 - 36 n^4 p q^2 -
18 n^4 p^2 q^2 + 24 n^5 p^3 q^2 - 20 n^3 q^3 - 12 n^4 q^3 -
4 n^4 p q^3 + 24 n^5 p^2 q^3 + 2 n^4 q^4 +
4 n^5 p q^4) #1^3 + (15 + 50 n + 48 n^2 + 15 n^3 + n^4 -
50 n p - 96 n^2 p - 45 n^3 p - 4 n^4 p + 45 n^2 p^2 +
45 n^3 p^2 + 6 n^4 p^2 - 10 n^3 p^3 - 4 n^4 p^3 - 50 n q -
96 n^2 q - 45 n^3 q - 4 n^4 q + 87 n^2 p q + 90 n^3 p q +
12 n^4 p q - 15 n^3 p^2 q - 12 n^4 p^2 q - 12 n^4 p^3 q +
45 n^2 q^2 + 45 n^3 q^2 + 6 n^4 q^2 - 15 n^3 p q^2 -
12 n^4 p q^2 - 30 n^4 p^2 q^2 - 10 n^3 q^3 - 4 n^4 q^3 -
12 n^4 p q^3) #1^4 + (20 + 45 n + 27 n^2 + 4 n^3 - 42 n p -
54 n^2 p - 12 n^3 p + 18 n^2 p^2 + 12 n^3 p^2 - 42 n q -
54 n^2 q - 12 n^3 q + 30 n^2 p q + 24 n^3 p q +
12 n^3 p^2 q + 18 n^2 q^2 + 12 n^3 q^2 +
12 n^3 p q^2) #1^5 + (14 + 21 n + 6 n^2 - 14 n p - 12 n^2 p -
14 n q - 12 n^2 q - 4 n^2 p q) #1^6 + (4 + 4 n) #1^7 &, 2] ||
l == Root[-n^6 p^3 q^3 + n^7 p^4 q^3 + n^7 p^3 q^4 -
n^8 p^4 q^4 + (n + 3 n^2 + 3 n^3 + n^4 - n p - 6 n^2 p -
9 n^3 p - 4 n^4 p + 3 n^2 p^2 + 9 n^3 p^2 + 6 n^4 p^2 -
3 n^3 p^3 - 4 n^4 p^3 + n^4 p^4 - n q - 6 n^2 q - 9 n^3 q -
4 n^4 q + 6 n^2 p q + 18 n^3 p q + 12 n^4 p q - 9 n^3 p^2 q -
12 n^4 p^2 q + 4 n^4 p^3 q + 3 n^2 q^2 + 9 n^3 q^2 +
6 n^4 q^2 - 9 n^3 p q^2 - 12 n^4 p q^2 + 6 n^4 p^2 q^2 +
3 n^5 p^3 q^2 - 3 n^6 p^4 q^2 - 3 n^3 q^3 - 4 n^4 q^3 +
4 n^4 p q^3 + 3 n^5 p^2 q^3 - 8 n^6 p^3 q^3 + 4 n^7 p^4 q^3 +
n^4 q^4 - 3 n^6 p^2 q^4 + 4 n^7 p^3 q^4) #1 + (1 + 9 n +
18 n^2 + 13 n^3 + 3 n^4 - 9 n p - 36 n^2 p - 39 n^3 p -
12 n^4 p + 18 n^2 p^2 + 39 n^3 p^2 + 18 n^4 p^2 -
13 n^3 p^3 - 12 n^4 p^3 + 3 n^4 p^4 - 9 n q - 36 n^2 q -
39 n^3 q - 12 n^4 q + 36 n^2 p q + 78 n^3 p q + 36 n^4 p q -
39 n^3 p^2 q - 36 n^4 p^2 q + 9 n^4 p^3 q + 3 n^5 p^4 q +
18 n^2 q^2 + 39 n^3 q^2 + 18 n^4 q^2 - 39 n^3 p q^2 -
36 n^4 p q^2 + 9 n^4 p^2 q^2 + 18 n^5 p^3 q^2 -
6 n^6 p^4 q^2 - 13 n^3 q^3 - 12 n^4 q^3 + 9 n^4 p q^3 +
18 n^5 p^2 q^3 - 16 n^6 p^3 q^3 + 3 n^4 q^4 + 3 n^5 p q^4 -
6 n^6 p^2 q^4) #1^2 + (6 + 30 n + 42 n^2 + 21 n^3 + 3 n^4 -
30 n p - 84 n^2 p - 63 n^3 p - 12 n^4 p + 42 n^2 p^2 +
63 n^3 p^2 + 18 n^4 p^2 - 20 n^3 p^3 - 12 n^4 p^3 +
2 n^4 p^4 - 30 n q - 84 n^2 q - 63 n^3 q - 12 n^4 q +
84 n^2 p q + 126 n^3 p q + 36 n^4 p q - 54 n^3 p^2 q -
36 n^4 p^2 q - 4 n^4 p^3 q + 4 n^5 p^4 q + 42 n^2 q^2 +
63 n^3 q^2 + 18 n^4 q^2 - 54 n^3 p q^2 - 36 n^4 p q^2 -
18 n^4 p^2 q^2 + 24 n^5 p^3 q^2 - 20 n^3 q^3 - 12 n^4 q^3 -
4 n^4 p q^3 + 24 n^5 p^2 q^3 + 2 n^4 q^4 +
4 n^5 p q^4) #1^3 + (15 + 50 n + 48 n^2 + 15 n^3 + n^4 -
50 n p - 96 n^2 p - 45 n^3 p - 4 n^4 p + 45 n^2 p^2 +
45 n^3 p^2 + 6 n^4 p^2 - 10 n^3 p^3 - 4 n^4 p^3 - 50 n q -
96 n^2 q - 45 n^3 q - 4 n^4 q + 87 n^2 p q + 90 n^3 p q +
12 n^4 p q - 15 n^3 p^2 q - 12 n^4 p^2 q - 12 n^4 p^3 q +
45 n^2 q^2 + 45 n^3 q^2 + 6 n^4 q^2 - 15 n^3 p q^2 -
12 n^4 p q^2 - 30 n^4 p^2 q^2 - 10 n^3 q^3 - 4 n^4 q^3 -
12 n^4 p q^3) #1^4 + (20 + 45 n + 27 n^2 + 4 n^3 - 42 n p -
54 n^2 p - 12 n^3 p + 18 n^2 p^2 + 12 n^3 p^2 - 42 n q -
54 n^2 q - 12 n^3 q + 30 n^2 p q + 24 n^3 p q +
12 n^3 p^2 q + 18 n^2 q^2 + 12 n^3 q^2 +
12 n^3 p q^2) #1^5 + (14 + 21 n + 6 n^2 - 14 n p - 12 n^2 p -
14 n q - 12 n^2 q - 4 n^2 p q) #1^6 + (4 + 4 n) #1^7 &, 3] ||
l == Root[-n^6 p^3 q^3 + n^7 p^4 q^3 + n^7 p^3 q^4 -
n^8 p^4 q^4 + (n + 3 n^2 + 3 n^3 + n^4 - n p - 6 n^2 p -
9 n^3 p - 4 n^4 p + 3 n^2 p^2 + 9 n^3 p^2 + 6 n^4 p^2 -
3 n^3 p^3 - 4 n^4 p^3 + n^4 p^4 - n q - 6 n^2 q - 9 n^3 q -
4 n^4 q + 6 n^2 p q + 18 n^3 p q + 12 n^4 p q - 9 n^3 p^2 q -
12 n^4 p^2 q + 4 n^4 p^3 q + 3 n^2 q^2 + 9 n^3 q^2 +
6 n^4 q^2 - 9 n^3 p q^2 - 12 n^4 p q^2 + 6 n^4 p^2 q^2 +
3 n^5 p^3 q^2 - 3 n^6 p^4 q^2 - 3 n^3 q^3 - 4 n^4 q^3 +
4 n^4 p q^3 + 3 n^5 p^2 q^3 - 8 n^6 p^3 q^3 + 4 n^7 p^4 q^3 +
n^4 q^4 - 3 n^6 p^2 q^4 + 4 n^7 p^3 q^4) #1 + (1 + 9 n +
18 n^2 + 13 n^3 + 3 n^4 - 9 n p - 36 n^2 p - 39 n^3 p -
12 n^4 p + 18 n^2 p^2 + 39 n^3 p^2 + 18 n^4 p^2 -
13 n^3 p^3 - 12 n^4 p^3 + 3 n^4 p^4 - 9 n q - 36 n^2 q -
39 n^3 q - 12 n^4 q + 36 n^2 p q + 78 n^3 p q + 36 n^4 p q -
39 n^3 p^2 q - 36 n^4 p^2 q + 9 n^4 p^3 q + 3 n^5 p^4 q +
18 n^2 q^2 + 39 n^3 q^2 + 18 n^4 q^2 - 39 n^3 p q^2 -
36 n^4 p q^2 + 9 n^4 p^2 q^2 + 18 n^5 p^3 q^2 -
6 n^6 p^4 q^2 - 13 n^3 q^3 - 12 n^4 q^3 + 9 n^4 p q^3 +
18 n^5 p^2 q^3 - 16 n^6 p^3 q^3 + 3 n^4 q^4 + 3 n^5 p q^4 -
6 n^6 p^2 q^4) #1^2 + (6 + 30 n + 42 n^2 + 21 n^3 + 3 n^4 -
30 n p - 84 n^2 p - 63 n^3 p - 12 n^4 p + 42 n^2 p^2 +
63 n^3 p^2 + 18 n^4 p^2 - 20 n^3 p^3 - 12 n^4 p^3 +
2 n^4 p^4 - 30 n q - 84 n^2 q - 63 n^3 q - 12 n^4 q +
84 n^2 p q + 126 n^3 p q + 36 n^4 p q - 54 n^3 p^2 q -
36 n^4 p^2 q - 4 n^4 p^3 q + 4 n^5 p^4 q + 42 n^2 q^2 +
63 n^3 q^2 + 18 n^4 q^2 - 54 n^3 p q^2 - 36 n^4 p q^2 -
18 n^4 p^2 q^2 + 24 n^5 p^3 q^2 - 20 n^3 q^3 - 12 n^4 q^3 -
4 n^4 p q^3 + 24 n^5 p^2 q^3 + 2 n^4 q^4 +
4 n^5 p q^4) #1^3 + (15 + 50 n + 48 n^2 + 15 n^3 + n^4 -
50 n p - 96 n^2 p - 45 n^3 p - 4 n^4 p + 45 n^2 p^2 +
45 n^3 p^2 + 6 n^4 p^2 - 10 n^3 p^3 - 4 n^4 p^3 - 50 n q -
96 n^2 q - 45 n^3 q - 4 n^4 q + 87 n^2 p q + 90 n^3 p q +
12 n^4 p q - 15 n^3 p^2 q - 12 n^4 p^2 q - 12 n^4 p^3 q +
45 n^2 q^2 + 45 n^3 q^2 + 6 n^4 q^2 - 15 n^3 p q^2 -
12 n^4 p q^2 - 30 n^4 p^2 q^2 - 10 n^3 q^3 - 4 n^4 q^3 -
12 n^4 p q^3) #1^4 + (20 + 45 n + 27 n^2 + 4 n^3 - 42 n p -
54 n^2 p - 12 n^3 p + 18 n^2 p^2 + 12 n^3 p^2 - 42 n q -
54 n^2 q - 12 n^3 q + 30 n^2 p q + 24 n^3 p q +
12 n^3 p^2 q + 18 n^2 q^2 + 12 n^3 q^2 +
12 n^3 p q^2) #1^5 + (14 + 21 n + 6 n^2 - 14 n p - 12 n^2 p -
14 n q - 12 n^2 q - 4 n^2 p q) #1^6 + (4 + 4 n) #1^7 &, 4] ||
l == Root[-n^6 p^3 q^3 + n^7 p^4 q^3 + n^7 p^3 q^4 -
n^8 p^4 q^4 + (n + 3 n^2 + 3 n^3 + n^4 - n p - 6 n^2 p -
9 n^3 p - 4 n^4 p + 3 n^2 p^2 + 9 n^3 p^2 + 6 n^4 p^2 -
3 n^3 p^3 - 4 n^4 p^3 + n^4 p^4 - n q - 6 n^2 q - 9 n^3 q -
4 n^4 q + 6 n^2 p q + 18 n^3 p q + 12 n^4 p q - 9 n^3 p^2 q -
12 n^4 p^2 q + 4 n^4 p^3 q + 3 n^2 q^2 + 9 n^3 q^2 +
6 n^4 q^2 - 9 n^3 p q^2 - 12 n^4 p q^2 + 6 n^4 p^2 q^2 +
3 n^5 p^3 q^2 - 3 n^6 p^4 q^2 - 3 n^3 q^3 - 4 n^4 q^3 +
4 n^4 p q^3 + 3 n^5 p^2 q^3 - 8 n^6 p^3 q^3 + 4 n^7 p^4 q^3 +
n^4 q^4 - 3 n^6 p^2 q^4 + 4 n^7 p^3 q^4) #1 + (1 + 9 n +
18 n^2 + 13 n^3 + 3 n^4 - 9 n p - 36 n^2 p - 39 n^3 p -
12 n^4 p + 18 n^2 p^2 + 39 n^3 p^2 + 18 n^4 p^2 -
13 n^3 p^3 - 12 n^4 p^3 + 3 n^4 p^4 - 9 n q - 36 n^2 q -
39 n^3 q - 12 n^4 q + 36 n^2 p q + 78 n^3 p q + 36 n^4 p q -
39 n^3 p^2 q - 36 n^4 p^2 q + 9 n^4 p^3 q + 3 n^5 p^4 q +
18 n^2 q^2 + 39 n^3 q^2 + 18 n^4 q^2 - 39 n^3 p q^2 -
36 n^4 p q^2 + 9 n^4 p^2 q^2 + 18 n^5 p^3 q^2 -
6 n^6 p^4 q^2 - 13 n^3 q^3 - 12 n^4 q^3 + 9 n^4 p q^3 +
18 n^5 p^2 q^3 - 16 n^6 p^3 q^3 + 3 n^4 q^4 + 3 n^5 p q^4 -
6 n^6 p^2 q^4) #1^2 + (6 + 30 n + 42 n^2 + 21 n^3 + 3 n^4 -
30 n p - 84 n^2 p - 63 n^3 p - 12 n^4 p + 42 n^2 p^2 +
63 n^3 p^2 + 18 n^4 p^2 - 20 n^3 p^3 - 12 n^4 p^3 +
2 n^4 p^4 - 30 n q - 84 n^2 q - 63 n^3 q - 12 n^4 q +
84 n^2 p q + 126 n^3 p q + 36 n^4 p q - 54 n^3 p^2 q -
36 n^4 p^2 q - 4 n^4 p^3 q + 4 n^5 p^4 q + 42 n^2 q^2 +
63 n^3 q^2 + 18 n^4 q^2 - 54 n^3 p q^2 - 36 n^4 p q^2 -
18 n^4 p^2 q^2 + 24 n^5 p^3 q^2 - 20 n^3 q^3 - 12 n^4 q^3 -
4 n^4 p q^3 + 24 n^5 p^2 q^3 + 2 n^4 q^4 +
4 n^5 p q^4) #1^3 + (15 + 50 n + 48 n^2 + 15 n^3 + n^4 -
50 n p - 96 n^2 p - 45 n^3 p - 4 n^4 p + 45 n^2 p^2 +
45 n^3 p^2 + 6 n^4 p^2 - 10 n^3 p^3 - 4 n^4 p^3 - 50 n q -
96 n^2 q - 45 n^3 q - 4 n^4 q + 87 n^2 p q + 90 n^3 p q +
12 n^4 p q - 15 n^3 p^2 q - 12 n^4 p^2 q - 12 n^4 p^3 q +
45 n^2 q^2 + 45 n^3 q^2 + 6 n^4 q^2 - 15 n^3 p q^2 -
12 n^4 p q^2 - 30 n^4 p^2 q^2 - 10 n^3 q^3 - 4 n^4 q^3 -
12 n^4 p q^3) #1^4 + (20 + 45 n + 27 n^2 + 4 n^3 - 42 n p -
54 n^2 p - 12 n^3 p + 18 n^2 p^2 + 12 n^3 p^2 - 42 n q -
54 n^2 q - 12 n^3 q + 30 n^2 p q + 24 n^3 p q +
12 n^3 p^2 q + 18 n^2 q^2 + 12 n^3 q^2 +
12 n^3 p q^2) #1^5 + (14 + 21 n + 6 n^2 - 14 n p - 12 n^2 p -
14 n q - 12 n^2 q - 4 n^2 p q) #1^6 + (4 + 4 n) #1^7 &, 5] ||
l == Root[-n^6 p^3 q^3 + n^7 p^4 q^3 + n^7 p^3 q^4 -
n^8 p^4 q^4 + (n + 3 n^2 + 3 n^3 + n^4 - n p - 6 n^2 p -
9 n^3 p - 4 n^4 p + 3 n^2 p^2 + 9 n^3 p^2 + 6 n^4 p^2 -
3 n^3 p^3 - 4 n^4 p^3 + n^4 p^4 - n q - 6 n^2 q - 9 n^3 q -
4 n^4 q + 6 n^2 p q + 18 n^3 p q + 12 n^4 p q - 9 n^3 p^2 q -
12 n^4 p^2 q + 4 n^4 p^3 q + 3 n^2 q^2 + 9 n^3 q^2 +
6 n^4 q^2 - 9 n^3 p q^2 - 12 n^4 p q^2 + 6 n^4 p^2 q^2 +
3 n^5 p^3 q^2 - 3 n^6 p^4 q^2 - 3 n^3 q^3 - 4 n^4 q^3 +
4 n^4 p q^3 + 3 n^5 p^2 q^3 - 8 n^6 p^3 q^3 + 4 n^7 p^4 q^3 +
n^4 q^4 - 3 n^6 p^2 q^4 + 4 n^7 p^3 q^4) #1 + (1 + 9 n +
18 n^2 + 13 n^3 + 3 n^4 - 9 n p - 36 n^2 p - 39 n^3 p -
12 n^4 p + 18 n^2 p^2 + 39 n^3 p^2 + 18 n^4 p^2 -
13 n^3 p^3 - 12 n^4 p^3 + 3 n^4 p^4 - 9 n q - 36 n^2 q -
39 n^3 q - 12 n^4 q + 36 n^2 p q + 78 n^3 p q + 36 n^4 p q -
39 n^3 p^2 q - 36 n^4 p^2 q + 9 n^4 p^3 q + 3 n^5 p^4 q +
18 n^2 q^2 + 39 n^3 q^2 + 18 n^4 q^2 - 39 n^3 p q^2 -
36 n^4 p q^2 + 9 n^4 p^2 q^2 + 18 n^5 p^3 q^2 -
6 n^6 p^4 q^2 - 13 n^3 q^3 - 12 n^4 q^3 + 9 n^4 p q^3 +
18 n^5 p^2 q^3 - 16 n^6 p^3 q^3 + 3 n^4 q^4 + 3 n^5 p q^4 -
6 n^6 p^2 q^4) #1^2 + (6 + 30 n + 42 n^2 + 21 n^3 + 3 n^4 -
30 n p - 84 n^2 p - 63 n^3 p - 12 n^4 p + 42 n^2 p^2 +
63 n^3 p^2 + 18 n^4 p^2 - 20 n^3 p^3 - 12 n^4 p^3 +
2 n^4 p^4 - 30 n q - 84 n^2 q - 63 n^3 q - 12 n^4 q +
84 n^2 p q + 126 n^3 p q + 36 n^4 p q - 54 n^3 p^2 q -
36 n^4 p^2 q - 4 n^4 p^3 q + 4 n^5 p^4 q + 42 n^2 q^2 +
63 n^3 q^2 + 18 n^4 q^2 - 54 n^3 p q^2 - 36 n^4 p q^2 -
18 n^4 p^2 q^2 + 24 n^5 p^3 q^2 - 20 n^3 q^3 - 12 n^4 q^3 -
4 n^4 p q^3 + 24 n^5 p^2 q^3 + 2 n^4 q^4 +
4 n^5 p q^4) #1^3 + (15 + 50 n + 48 n^2 + 15 n^3 + n^4 -
50 n p - 96 n^2 p - 45 n^3 p - 4 n^4 p + 45 n^2 p^2 +
45 n^3 p^2 + 6 n^4 p^2 - 10 n^3 p^3 - 4 n^4 p^3 - 50 n q -
96 n^2 q - 45 n^3 q - 4 n^4 q + 87 n^2 p q + 90 n^3 p q +
12 n^4 p q - 15 n^3 p^2 q - 12 n^4 p^2 q - 12 n^4 p^3 q +
45 n^2 q^2 + 45 n^3 q^2 + 6 n^4 q^2 - 15 n^3 p q^2 -
12 n^4 p q^2 - 30 n^4 p^2 q^2 - 10 n^3 q^3 - 4 n^4 q^3 -
12 n^4 p q^3) #1^4 + (20 + 45 n + 27 n^2 + 4 n^3 - 42 n p -
54 n^2 p - 12 n^3 p + 18 n^2 p^2 + 12 n^3 p^2 - 42 n q -
54 n^2 q - 12 n^3 q + 30 n^2 p q + 24 n^3 p q +
12 n^3 p^2 q + 18 n^2 q^2 + 12 n^3 q^2 +
12 n^3 p q^2) #1^5 + (14 + 21 n + 6 n^2 - 14 n p - 12 n^2 p -
14 n q - 12 n^2 q - 4 n^2 p q) #1^6 + (4 + 4 n) #1^7 &, 6] ||
l == Root[-n^6 p^3 q^3 + n^7 p^4 q^3 + n^7 p^3 q^4 -
n^8 p^4 q^4 + (n + 3 n^2 + 3 n^3 + n^4 - n p - 6 n^2 p -
9 n^3 p - 4 n^4 p + 3 n^2 p^2 + 9 n^3 p^2 + 6 n^4 p^2 -
3 n^3 p^3 - 4 n^4 p^3 + n^4 p^4 - n q - 6 n^2 q - 9 n^3 q -
4 n^4 q + 6 n^2 p q + 18 n^3 p q + 12 n^4 p q - 9 n^3 p^2 q -
12 n^4 p^2 q + 4 n^4 p^3 q + 3 n^2 q^2 + 9 n^3 q^2 +
6 n^4 q^2 - 9 n^3 p q^2 - 12 n^4 p q^2 + 6 n^4 p^2 q^2 +
3 n^5 p^3 q^2 - 3 n^6 p^4 q^2 - 3 n^3 q^3 - 4 n^4 q^3 +
4 n^4 p q^3 + 3 n^5 p^2 q^3 - 8 n^6 p^3 q^3 + 4 n^7 p^4 q^3 +
n^4 q^4 - 3 n^6 p^2 q^4 + 4 n^7 p^3 q^4) #1 + (1 + 9 n +
18 n^2 + 13 n^3 + 3 n^4 - 9 n p - 36 n^2 p - 39 n^3 p -
12 n^4 p + 18 n^2 p^2 + 39 n^3 p^2 + 18 n^4 p^2 -
13 n^3 p^3 - 12 n^4 p^3 + 3 n^4 p^4 - 9 n q - 36 n^2 q -
39 n^3 q - 12 n^4 q + 36 n^2 p q + 78 n^3 p q + 36 n^4 p q -
39 n^3 p^2 q - 36 n^4 p^2 q + 9 n^4 p^3 q + 3 n^5 p^4 q +
18 n^2 q^2 + 39 n^3 q^2 + 18 n^4 q^2 - 39 n^3 p q^2 -
36 n^4 p q^2 + 9 n^4 p^2 q^2 + 18 n^5 p^3 q^2 -
6 n^6 p^4 q^2 - 13 n^3 q^3 - 12 n^4 q^3 + 9 n^4 p q^3 +
18 n^5 p^2 q^3 - 16 n^6 p^3 q^3 + 3 n^4 q^4 + 3 n^5 p q^4 -
6 n^6 p^2 q^4) #1^2 + (6 + 30 n + 42 n^2 + 21 n^3 + 3 n^4 -
30 n p - 84 n^2 p - 63 n^3 p - 12 n^4 p + 42 n^2 p^2 +
63 n^3 p^2 + 18 n^4 p^2 - 20 n^3 p^3 - 12 n^4 p^3 +
2 n^4 p^4 - 30 n q - 84 n^2 q - 63 n^3 q - 12 n^4 q +
84 n^2 p q + 126 n^3 p q + 36 n^4 p q - 54 n^3 p^2 q -
36 n^4 p^2 q - 4 n^4 p^3 q + 4 n^5 p^4 q + 42 n^2 q^2 +
63 n^3 q^2 + 18 n^4 q^2 - 54 n^3 p q^2 - 36 n^4 p q^2 -
18 n^4 p^2 q^2 + 24 n^5 p^3 q^2 - 20 n^3 q^3 - 12 n^4 q^3 -
4 n^4 p q^3 + 24 n^5 p^2 q^3 + 2 n^4 q^4 +
4 n^5 p q^4) #1^3 + (15 + 50 n + 48 n^2 + 15 n^3 + n^4 -
50 n p - 96 n^2 p - 45 n^3 p - 4 n^4 p + 45 n^2 p^2 +
45 n^3 p^2 + 6 n^4 p^2 - 10 n^3 p^3 - 4 n^4 p^3 - 50 n q -
96 n^2 q - 45 n^3 q - 4 n^4 q + 87 n^2 p q + 90 n^3 p q +
12 n^4 p q - 15 n^3 p^2 q - 12 n^4 p^2 q - 12 n^4 p^3 q +
45 n^2 q^2 + 45 n^3 q^2 + 6 n^4 q^2 - 15 n^3 p q^2 -
12 n^4 p q^2 - 30 n^4 p^2 q^2 - 10 n^3 q^3 - 4 n^4 q^3 -
12 n^4 p q^3) #1^4 + (20 + 45 n + 27 n^2 + 4 n^3 - 42 n p -
54 n^2 p - 12 n^3 p + 18 n^2 p^2 + 12 n^3 p^2 - 42 n q -
54 n^2 q - 12 n^3 q + 30 n^2 p q + 24 n^3 p q +
12 n^3 p^2 q + 18 n^2 q^2 + 12 n^3 q^2 +
12 n^3 p q^2) #1^5 + (14 + 21 n + 6 n^2 - 14 n p - 12 n^2 p -
14 n q - 12 n^2 q - 4 n^2 p q) #1^6 + (4 + 4 n) #1^7 &, 7]


is a set of equations for l like in the question but not intuitive or illustrative.

Most users with little knowledge or experience are attracted by the built-ins with solve. That is not he real power Mathematica hid behind this group of metacognitive built-in names. Polynomials are in general nicer solved with the roots-group of built-ins.