Solving the equation:
x3-300x=2961 was one of the problems in Oxford entrance requirements.
A solution published on the web is based on calculus which was not to be assumed as possessed by all applicants.
I ‘ve solved it p&p, and almost in no time.
Request to provide a KISS (sans calculus) solution.
x^3 - 300*x - 2961 = 0
Try the rational root theorem.
The prime factors of 2961 are 3,3,7,47.
So if there are integer roots, they are limited to (+ or -):
1, 3, 7, 9, 21, 47, 63, 141, 329, 423, 987, 2961
From Descartes' rule of signs, there is one positive real root and either 0 or 2 negative real roots.
Since x^3 is larger than 2961, the positive root has to be larger than the cube root of 2961; so it has to be larger than 14 or 15. So let's try 21.
21 works. So x^3 - 300*x - 2961 = (x-21)*(a*x^2 + b*x + c)
The coefficient of x^2, a, has to be 1.
c has to be (-2961) / (-21) = +141
And since x^3 - 300*x - 2961 has no x^2 term, b must be -(-21) = +21.
x^3 - 300*x - 2961 = (x-21)*(x^2 + 21x + 141)
The only real root is 21.
The other 2 roots are irrational: [-21 +/- sqrt(123)i]/2
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Posted by Larry
on 2022-12-13 09:34:33 |
One way
