Find roots to quintic polynomial

In summary, the student is trying to solve for the roots of a polynomial x4+x+1. They are using tricks from their algebra class to find the roots, but they are not sure if there are any real roots. If there are no real roots, then they will need to multiple the conjugate pairs to find all of the solutions.
  • #1
VeeEight
615
0

Homework Statement


Find the roots to the polynomial x4+x+1

this is a problem I am stuck with for my algebra class

Homework Equations


there are a few tricks I have learned for finding roots like taking conjugates and rewriting using fundamental theorem of algebra but we haven't worked on this stuff too much


The Attempt at a Solution



by fundamental theorem of algebra, we can write x4+x+1 = (x-a)(x-b)(x-c)(x-d) for some complex numbers a,b,c,d

so the roots a,b,c,d have to follow the equations:
a + b + c + d = 0
ad + ab + bd + cd + bc = 0
abd + bdc + acd + abc = 1
abcd = 1

other than that I do not know what else to do to find the roots
 
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  • #2
Any complex numbers that are solutions show up in pairs as conjugates a + bi and a - bi.

Also, this is a quartic (degree 4), not a quintic (degree 5).
 
  • #3
Mark44 said:
Any complex numbers that are solutions show up in pairs as conjugates a + bi and a - bi.

Also, this is a quartic (degree 4), not a quintic (degree 5).

Yup the part about conjugates is one of the only things I know about this subject
and thanks, I put down quintic when I should have said quartic.
 
  • #4
have a look at your equation, can you convince yourself there are no real roots? (ie is f(x) > 0 for all x in the reals?

if so as Mark implied, you know there will be 2 sets of complex conjugate roots, try inputting these in your equation and solving
 
  • #5
okay so the roots are of the form: r1 = a+ib, r2 = a-ib, r3 = c+id, r4 = c-id where a,b,c,d are reals.

then we obtain the equations 2a + 2c = 0 and
(a2+b2)(c2+d2) = 1
 
  • #6
on the right track, but you should get an equation for each coefficient of x
 
  • #7
okay so the equations are:
2a + 2c = 0
(a2 + b2) (c2 + d2) = 1
2(a2c + ac2 + ad2+b2c) + i (a2b + d2b + c2b+b2d + a2d + c2d + d3 + b3) = 0
a2 + b2 + c2 + d2 + 3ac + bc -i (ad + ac) = 1
 
  • #8
multiple the conjugate pairs first

for a complex number z = a + ib
z.z* = (a^2 + b^2) is real
z+z* = 2a is real
so (x-z)(x-z*) = x^2 - (z+z*)x + z.z* is then real

so there should be no coefficents of i in your equations, and there should be no d3 as it only appears in factors twice...
 
  • #9
I always seem to get coefficients of i in the equations though
 
  • #10
try this
for
z = a + ib, z* = a-ib
y = c + id, y* = c-id

(x-z)(x-z*)(x-y)(x-y*) = (x^2 - (z+z*)x + z.z*)(x^2 - (y+y*)x + y.y*)

there is clearly no i's left on the right hand side
 
Last edited:

What is a quintic polynomial?

A quintic polynomial is a type of polynomial expression with a degree of 5. This means it contains terms with variables raised to the 5th power, such as x^5.

Why is it important to find roots to quintic polynomials?

Finding the roots, or solutions, to a quintic polynomial allows us to solve equations and understand the behavior of the polynomial. It also has applications in fields such as engineering, physics, and economics.

How do you find the roots to a quintic polynomial?

There is no algebraic formula or method for finding the roots to a general quintic polynomial. However, we can use numerical methods such as Newton's method or the Durand-Kerner method to approximate the roots.

What is the complex root theorem for quintic polynomials?

The complex root theorem states that a quintic polynomial with real coefficients has either 5 real roots or 3 real roots and 1 complex conjugate pair of roots. This theorem can be useful in determining the number of real solutions to a quintic polynomial.

Can quintic polynomials always be solved exactly?

No, there are some quintic polynomials that cannot be solved exactly. This was proven by the Abel-Ruffini theorem, which states that there is no general formula for finding the roots of a quintic polynomial using only radicals and arithmetic operations. Therefore, numerical methods must be used to approximate the roots.

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