x is a whole number, x E Z
two solutions
look at the factors of 144
subtract 144 and factor the left (to hunt down the other solutions)
method 1:
144's factors are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
the numbers must have gaps of 1, 1, 1, 2 (x-2)(x-1)(x)(x+1)(x+3)
so it must be 3
method 2:
x(x-1)(x+1)(x-2)(x+3) = 144
x^5+x^4-7x^3-x^2+6x-144=0
the rational root theorem tells us that any rational roots must be factors of 144
we already found one up there with method 1
so x-3 must be a factor
(x-3)(x^4+4x^3+5x^2+14x+48)=0
lets look for other solutions with the rational root theorem
factors of 48 = +-(1, 2, 3, 4, 6, 8, 12, 16, 24, 48)
since all the coefficients are positive we know the other roots must be negative
i'm gonna cheat and use a graph, it turns out that none of the real numbers work
to the complex world it is then.
(someone help me on that part)
x=3
two solutions
look at the factors of 144
subtract 144 and factor the left (to hunt down the other solutions)
method 1:
144's factors are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
the numbers must have gaps of 1, 1, 1, 2 (x-2)(x-1)(x)(x+1)(x+3)
so it must be 3
method 2:
x(x-1)(x+1)(x-2)(x+3) = 144
x^5+x^4-7x^3-x^2+6x-144=0
the rational root theorem tells us that any rational roots must be factors of 144
we already found one up there with method 1
so x-3 must be a factor
(x-3)(x^4+4x^3+5x^2+14x+48)=0
lets look for other solutions with the rational root theorem
factors of 48 = +-(1, 2, 3, 4, 6, 8, 12, 16, 24, 48)
since all the coefficients are positive we know the other roots must be negative
i'm gonna cheat and use a graph, it turns out that none of the real numbers work
to the complex world it is then.
(someone help me on that part)
x=3