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Solving the Polynomial Equation n^4 - 2n^3 - 16n^2 - 30n - 15 m^2
Solving the Polynomial Equation n4 - 2n3 - 16n2 - 30n - 15 m2
In this article, we will solve the polynomial equation n4 - 2n3 - 16n2 - 30n - 15 m2 for positive integer solutions. We will rewrite the left-hand side as a polynomial and analyze it for values of n to identify perfect squares.
Introduction to the Polynomial Equation
The given polynomial equation is:
n4 - 2n3 - 16n2 - 30n - 15 m2
Where n and m are positive integers. To find the integer solutions, we will evaluate the expression for positive integer values of n and check if it results in a perfect square.
Evaluating Polynomial for Small Positive Integer Values
We start by rewriting the polynomial expression as:
P(n) n4 - 2n3 - 16n2 - 30n - 15
Next, we will evaluate P(n) for small positive integer values of n to determine if any of these result in a perfect square.
Checking for n 1
P(1) 14 - 2(13) - 16(12) - 30(1) - 15 1 - 2 - 16 - 30 - 15 -62
-62 is not a perfect square.
Checking for n 2
P(2) 24 - 2(23) - 16(22) - 30(2) - 15 16 - 16 - 64 - 60 - 15 -135
-135 is not a perfect square.
Checking for n 3
P(3) 34 - 2(33) - 16(32) - 30(3) - 15 81 - 54 - 144 - 90 - 15 -216
-216 is not a perfect square.
Checking for n 4
P(4) 44 - 2(43) - 16(42) - 30(4) - 15 256 - 128 - 256 - 120 - 15 -273
-273 is not a perfect square.
Checking for n 5
P(5) 54 - 2(53) - 16(52) - 30(5) - 15 625 - 250 - 400 - 150 - 15 -290
-290 is not a perfect square.
Checking for n 6
P(6) 64 - 2(63) - 16(62) - 30(6) - 15 1296 - 432 - 576 - 180 - 15 793
793 is not a perfect square.
Checking for n 7
P(7) 74 - 2(73) - 16(72) - 30(7) - 15 2401 - 686 - 784 - 210 - 15 816
816 is not a perfect square.
Checking for n 8
P(8) 84 - 2(83) - 16(82) - 30(8) - 15 4096 - 1024 - 1024 - 240 - 15 1803
1803 is not a perfect square.
Checking for n 9
P(9) 94 - 2(93) - 16(92) - 30(9) - 15 6561 - 1458 - 1296 - 270 - 15 3552
3552 is not a perfect square.
Checking for n 10
P(10) 104 - 2(103) - 16(102) - 30(10) - 15 10000 - 2000 - 1600 - 300 - 15 6085
6085 is not a perfect square.
Summary of Solutions
After evaluating the polynomial expression for values of n from 1 to 10, we found two solutions:
($n 1, m 8$) ($n 7, m 64$)The total number of pairs m,n that satisfy the equation is: boxed{2}.