How do you use the rational root theorem to list all possible roots for x^3+5x^2-2x-15=0x3+5x2−2x−15=0?
1 Answer
The "possible" rational roots are
The actual roots are:
x_k = 1/3(2sqrt(31) cos(1/3 cos^(-1)((65sqrt(31)) / 1922)+(2kpi)/3)-5)xk=13(2√31cos(13cos−1(65√311922)+2kπ3)−5)
Explanation:
Given:
f(x) = x^3+5x^2-2x-15f(x)=x3+5x2−2x−15
Rational root theorem
The rational root theorem will only tell you about possible rational roots - not irrational or Complex ones.
By the rational root theorem, any rational zeros of
That means that the only possible rational roots are:
+-1, +-3, +-5, +-15±1,±3,±5,±15
However, note that all of these are odd, resulting in all of the terms of
f(x) = "odd" + "odd" + "even" + "odd" = "odd" != 0f(x)=odd+odd+even+odd=odd≠0
So
Bonus
Let us see what else we can find out about
Discriminant
The discriminant
Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd
In our example,
Delta = 100+32+7500-6075+2700 = 4257
Since
Descartes's Rule of Signs
The signs of the coefficients of
Tschirnhaus transformation
To make the task of solving the cubic simpler, we make the cubic simpler using a linear substitution known as a Tschirnhaus transformation.
0=27f(x)=27x^3+135x^2-54x-405
=(3x+5)^3-93(3x+5)-65
=t^3-93t-65
where
Trigonometric substitution
Look for a substitution of the form:
t = k cos theta
where
4 cos^3 theta - 3 cos theta = cos 3 theta
Let
Then:
0 = t^3-93t-65
color(white)(0) = k^3cos^3 theta - 93k cos theta - 65
color(white)(0) = 2sqrt(31)(124 cos^3 theta - 93 cos theta) - 65
color(white)(0) = 62sqrt(31)(4 cos^3 theta - 3 cos theta) - 65
color(white)(0) = 62sqrt(31)cos 3theta - 65
Hence:
cos 3 theta = 65/(62 sqrt(31)) = (65sqrt(31)) / 1922
So:
3 theta = +-cos^(-1) ((65sqrt(31)) / 1922) + 2kpi
t = 2sqrt(31) cos(1/3 cos^(-1)((65sqrt(31)) / 1922)+(2kpi)/3)
Then
x_k = 1/3(2sqrt(31) cos(1/3 cos^(-1)((65sqrt(31)) / 1922)+(2kpi)/3)-5)