Solving quartic equations using synthetic division

[alingy1]

Homework Statement

We have a polynomial of fourth degree. I want to use synthetic division. However, is synthetic division applicable at all times? Are there cases in which the first binome extracted cannot have an integrer? What do I do at that time?

Homework Equations

Synthetic division.

The Attempt at a Solution

I have absolutely no clue which other method to use. I searched on purplemath and other websites, but can't find an example where the polynomials are fractions. I'm lost and I've been working on this for hours :P I'll take a break before reading your valuable answers. Thanks in advance.

 

[SteamKing]

Why don't you post your polynomial and remove a lot of the doubts and lack of clarity in your post.

What do you mean by 'where the polynomials are fractions'?

If synthetic division is to work, you must have found at least one root with which to use for the divisor of your original quartic. Have you done so?

In general, the following methods are applicable to finding solutions to quartics:
1. Determine a solution or solutions by trial and error.
2. Graph the quartic to determine where solutions are located.
3. Use the quartic formula (which is complicated): https://en.wikipedia.org/wiki/Quartic_function

Sometimes the following technique can be used:
4. If the form of the particular quartic permits, use substitution to reduce the degree of the quartic to a quadratic.

From algebra, quartics will have 4 real solutions or 2 real solutions and a pair of complex solutions, or 2 pairs of complex solutions.

 

[lurflurf]

Synthetic division will always work.
Often we are only considering roots of a certain form so we are limited by that
For example as above if we look for only rational roots and there are none we will not find any
That has nothing to do with synthetic division per se
If you want to factor a quartic and there are no rational roots you will have to use a method that finds the kind of roots that it has

 

[alingy1]

Why use the quartic formula when we could use synthetic division, which is much faster?

I don't have a specific example. Could you give me a case where there are no rational roots? I want to try solving it and see what the tell-tell signs of the fact that there is no solution are.

Using the factor theorem, purplemath.com says that there are cases where none of the values it gives for x=? at the beginning of factoring work. What does that mean? If there are only complex answers, how can I know it? What are those tell-tell signs (considering I don't have access to graphing calculators)?

"Note that I keep saying "potential" roots, "possible" zeroes, "if there are any such roots...". This is because the list of fractions generated by the Rational Roots Test is just a list of potential solutions. It need not be true that any of the fractions is actually a solution. There might not be any fractional roots! "
-http://www.purplemath.com/modules/rtnlroot.htm

I want to know if, when factoring a polynomial like that, it is okay to have answers with fractions:
i.e.:
(x+1,62351)(x-3,5151)(x+sqroot(4,45))(x+0,62)
(I made those answers up, but I hope you get the idea. In the textbook I use, I only see examples with integers. Is there any way to always get integers?)

 

[SteamKing]

Read post #2 carefully. In order to reduce the degree of the quartic polynomial using synthetic division, you must already know at least one of the solutions (roots, zeros, whatever) which you are attempting to find, otherwise, the synthetic division gives you a quotient term plus a remainder term, which indicates that the divisor originally used in the synthetic division is not a true factor to the original quartic polynomial.

Synthetic division is a simplified and modified form of the long division algorithm. It does not produce factors for a polynomial automatically.

The rational root theorem, Descartes rule of signs, and the fundamental theorem of algebra give us qualitative indications about the roots of a polynomial, but they do not always tell us what the value of the actual roots are. To determine the values of the roots, additional techniques must be used.

If you want to study polynomials in more detail, I recommend:

http://tutorial.math.lamar.edu/Classes/Alg/PolynomialFunctions.aspx

 

[lurflurf]

try something like
256 x^4-512 x^3+304 x^2-48 x^1+1=0
or
16 x^4+8 x^3-16 x^2-8 x+1
or
x^4-40 x^3+352 x^2-960 x^1+576

 

[HallsofIvy]

Why use the quartic formula when we could use synthetic division, which is much faster?

I am puzzled as to what you mean by "using synthetic division" to solve an equation!
"Synthetic division" is NOT a method for solving equations, it is a method for testing whether or not a given number is a solution. You first have to determine what numbers to try. How are you doing that?

 

[alingy1]

Well, there is only one way of finding the possible solution, the rational roots tests. Is there any other?

 

[alingy1]

Lurflurf, what are those answers? I want to know if it is possible to have fractions in the factored form. Those are simplified answers.

 

[Ray Vickson]

Well, there is only one way of finding the possible solution, the rational roots tests. Is there any other?

Yes: there are NUMERICAL methods. For polynomials of degree ≤ 4 there are formulas for the roots, but they are rarely used for cubic or quartic equations---the straight numerical methods are often faster and often more accurate when we take computational roundoff errors into account. For degree ≥ 5 there are NO formulas for the roots in terms of elementary operations, and in such cases we are forced (in general) to fall back on numerical methods, or do computations with hypergeometric functions.

 

[Ray Vickson]

Lurflurf, what are those answers? I want to know if it is possible to have fractions in the factored form. Those are simplified answers.

Submit the equations to Wolfram Alpha and see what you get.

 

[lurflurf]

My examples have square roots, they demonstrate that a quartic with integer coefficients can have no rational roots. As far as numerical methods it depends. As far as degree >=5 it is true there is no formula in terms of roots for all coefficients, but that is only a curiosity as often a formula in terms of roots for selected coefficients or a formula using operations other than roots is useful.

 

[alingy1]

Hmm, I'm starting to get lost again :S
I have this:
4x^3-6x^2+3x-2=0
I looked on wolfram. I get a zero like 1,2211.
How can I find that all by myself!?
I tried to factor it using the rational test method, which leads to nowhere since every possible answer is wrong.

 

[SteamKing]

That's because 1.2211 is an irrational number, i.e. it cannot be expressed as the ratio of two integers.
The rational root theorem is not guaranteed to lead to a root: it can only provide a set of possible roots which must then be verified by substitution into the original polynomial. The rule of signs says that the polynomial in post #13 has either three reals roots or one root: Mathematica has demonstrated that there is one real root and a pair of complex roots.

IMO, you are dabbling with polynomials without understanding what algebra says about the nature of their roots nor how to go about finding them in a systematic fashion. I recommend you study this website or one similar:
http://tutorial.math.lamar.edu/Classes/Alg/PolynomialFunctions.aspx

 

[alingy1]

Oh, that's what bummed me. Your website was very helpful. I read it before today too. But, I'm still confused in how we could find irrational roots. Is that possible?

 

[Ray Vickson]

Oh, that's what bummed me. Your website was very helpful. I read it before today too. But, I'm still confused in how we could find irrational roots. Is that possible?

It has already been explained to you that for quadratic, cubic and quartic equations there are formulas you can use to find the roots. No matter whether the roots are rational or irrational, the formulas will apply equally well. However, as has also been explained to you, we rarely use the formulas for cubic or quartic equations; most often, when the roots are irrational we just resort to numerical methods.

 

[lurflurf]

Hmm, I'm starting to get lost again :S
I have this:
4x^3-6x^2+3x-2=0
I looked on wolfram. I get a zero like 1,2211.
How can I find that all by myself!?
I tried to factor it using the rational test method, which leads to nowhere since every possible answer is wrong.

Again rational test method is very limited. When we have an equation like
4x^3-6x^2+3x-2=0
The first thing we usually do is a Tschirnhaus transformation
given
ax^3+bx^2+cx^3+d=0
we write x as ((x+b/3a)-b/3a)
as you can work out b/3a is chosen to eliminate the quadradic term, but here we fair even better.
4x^3-6x^2+3x-2=0
writing
((x+b/3a)-b/3a)=((x-1/2)+1/2)
(8(x-1/2)^3-3)/2=0
which is easy to solve

 

[Ray Vickson]

Oh, that's what bummed me. Your website was very helpful. I read it before today too. But, I'm still confused in how we could find irrational roots. Is that possible?

To further expand on the answer I gave before, here is why we normally do not try to use formulas to solve quartic equations, even though it can be done. For the equation
x^4+a*x^3+b*x^2+c*x+e=0, one of the solutions (obtained using the computer algebra package Maple) is
root1 = -1/4*a+1/12*3^(1/2)*((3*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)-8*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)+2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-24*c*a+96*e+8*b^2)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3))^(1/2)+1/12*(-(-18*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)*((3*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)-8*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)+2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-24*c*a+96*e+8*b^2)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3))^(1/2)+48*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)*((3*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)-8*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)+2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-24*c*a+96*e+8*b^2)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3))^(1/2)+6*((3*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)-8*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)+2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-24*c*a+96*e+8*b^2)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3))^(1/2)*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-72*((3*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)-8*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)+2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-24*c*a+96*e+8*b^2)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3))^(1/2)*c*a+288*((3*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)-8*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)+2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-24*c*a+96*e+8*b^2)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3))^(1/2)*e+24*((3*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)-8*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)+2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-24*c*a+96*e+8*b^2)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3))^(1/2)*b^2-72*3^(1/2)*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)*a*b+18*3^(1/2)*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)*a^3+144*3^(1/2)*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)*c)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)/((3*a^2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)-8*b*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3)+2*(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(2/3)-24*c*a+96*e+8*b^2)/(-36*b*c*a-288*e*b+108*c^2+108*e*a^2+8*b^3+12*(-3*c^2*a^2*b^2-54*b*c^3*a+18*c^2*a^2*e-432*e^2*b*a^2+240*c*a*e*b^2-54*b*c*a^3*e+12*e*a^2*b^3+384*e^2*b^2-48*e*b^4+12*c^2*b^3+81*e^2*a^4+12*c^3*a^3-768*e^3+81*c^4+576*c*a*e^2-432*e*b*c^2)^(1/2))^(1/3))^(1/2))^(1/2)

There are three other roots similar to this one that Maple has also found.

 

[Bill Simpson]

Well, there is only one way of finding the possible solution, the rational roots tests. Is there any other?

There is one other method that doesn't require that you already know the root. It only works when you have a repeated root, p=(x-a)^n*(x-b)... where n>1, but you don't need to know any of the a,b,... to use this.

Use calculus to find the derivative of p. Then find the greatest common denominator between the original polynomial and the derivative. The result will be 1 if there are no repeated roots and will be the repeated root or a product of possible powers of those roots if there are several roots to powers greater than 2. Like everything else you need to be careful to not make mistakes.

If your problem always has square roots then this method will always give you a root.

In[1]:= p = Expand[(x - 3)^2*(x - 4) (x + 1)]
Out[1]= -36 - 3 x + 23 x^2 - 9 x^3 + x^4

In[2]:= D[p, x] (*find the derivative*)
Out[2]= -3 + 46 x - 27 x^2 + 4 x^3

In[3]:= PolynomialGCD[-36-3x+23x^2-9x^3+x^4, -3+46x-27x^2+4x^3]
Out[3]= -3 + x