How do you use the rational root theorem to list all possible roots for x^3+5x^2-2x-15=0?

1 Answer
Nov 18, 2016

The "possible" rational roots are +-1, +-3, +-5, +-15

The actual roots are:

x_k = 1/3(2sqrt(31) cos(1/3 cos^(-1)((65sqrt(31)) / 1922)+(2kpi)/3)-5)

k = 0, 1, 2

Explanation:

Given:

f(x) = x^3+5x^2-2x-15

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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 f(x) are expressible in the form p/q for integers p, q with p a divisor of the constant term -15 and q a divisor of the coefficient 1 of the leading term.

That means that the only possible rational roots are:

+-1, +-3, +-5, +-15

However, note that all of these are odd, resulting in all of the terms of f(x) being odd, except -2x, which will be even. So for any of these values we have:

f(x) = "odd" + "odd" + "even" + "odd" = "odd" != 0

So f(x) has no rational zeros.

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Bonus

Let us see what else we can find out about f(x)'s zeros...

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Discriminant

The discriminant Delta of a cubic polynomial in the form ax^3+bx^2+cx+d is given by the formula:

Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd

In our example, a=1, b=5, c=-2 and d=-15, so we find:

Delta = 100+32+7500-6075+2700 = 4257

Since Delta > 0 this cubic has 3 Real zeros.

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Descartes's Rule of Signs

The signs of the coefficients of f(x) are in the pattern + + - -. With one change of sign, by Descartes' rule of signs, we can deduce that f(x) has exactly one positive Real zero. So the other two Real zeros are negative.

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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 t=(3x+5)

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Trigonometric substitution

Look for a substitution of the form:

t = k cos theta

where k is chosen such that the resulting cubic in cos theta contains:

4 cos^3 theta - 3 cos theta = cos 3 theta

Let k = 2sqrt(31)

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 = 1/3(t-5) and hence the zeros of our original cubic are:

x_k = 1/3(2sqrt(31) cos(1/3 cos^(-1)((65sqrt(31)) / 1922)+(2kpi)/3)-5)

k = 0, 1, 2