Roots of a Cubic Polynomial: Proving Coefficient Inequalities

In summary, the conversation discusses the proof that in the equation x^3+ax^2+bx+c=0, where the coefficients a,b and c are all real and the roots are all real and greater than 1, (i) a<-3, (ii) a^2>2b+3, and (iii) a^3<-9b-3c-3. The solution involves using the Vieta relations and manipulating the polynomial to prove each statement.
  • #1
rock.freak667
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Homework Statement


In the equation [tex]x^3+ax^2+bx+c=0[/tex]
the coefficients a,b and c are all real. It is given that all the roots are real and greater than 1.
(i) Prove that [tex]a<-3[/tex]
(ii)By considering the sum of the squares of the roots,prove that [tex]a^2>2b+3[/tex]
(iii)By considering the sum of the cubes of the roots,prove that [tex]a^3<-9b-3c-3[/tex]


Homework Equations



If the roots are A,B and C then A+B+C = a/1=a
ABC= -c/a
AB+AC+BC= b/a

The Attempt at a Solution



I do not know if there are any other formula for the squares/cubes of roots other than the ones i stated above; If there are any simpler ones please tell me.
I got out parts (ii) by taking (A+B+C)=a and appropriately squaring it, but I was unable to get out parts (i) and (iii), could someone please help me prove it..thanks
 
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  • #2
Since the roots are real we know the polynomial factors into [itex](x-r_1)(x-r_2)(x-r_3)[/itex]. Look at how [itex](x-r_1)(x-r_2)(x-r_3)[/itex] multiplies out and look at a b and c in terms of the roots. For example, we know that c must be negative as [itex]-r_1r_2r_3=c<0[/itex]. We actually know [itex]c<-1[/itex] as each of these roots are greater than 1.
 
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  • #3
thanks, I will re-try it and see now
 
  • #5
The point iii) is really tricky.

[tex] (A+B+C)^3 = A^3 +B^3 +C^3 -3ABC +3(A+B+C)(AB+AC+BC) [/tex]

which means

[tex] -a^3 =A^3 +B^3 +C^3 +3c -3ab > 3+3c+9b [/tex] ,

where i used the fact that the sum of the cubes is larger than 3 and the fact that a is smaller than -3.

Multiply by -1 and you're done.
 

1. What are the roots of a cubic polynomial?

The roots of a cubic polynomial are the solutions to the equation where the polynomial is equal to zero. In other words, the roots are the values of x that make the polynomial equal to zero.

2. How do you find the roots of a cubic polynomial?

To find the roots of a cubic polynomial, you can use the algebraic method of factoring or the numerical method of using a graphing calculator or computer software. The algebraic method involves factoring the polynomial into its linear and quadratic factors and solving for the roots. The numerical method involves using the polynomial function to graph it and finding the x-intercepts, or points where the polynomial crosses the x-axis, which are the roots.

3. Can a cubic polynomial have more than three roots?

No, a cubic polynomial can only have a maximum of three distinct roots. This is because a cubic polynomial is a third-degree polynomial, which means it has an exponent of 3 for its highest degree term. Since the fundamental theorem of algebra states that a polynomial of degree n can have at most n distinct roots, a cubic polynomial can have at most three roots.

4. What do the roots of a cubic polynomial tell us about the polynomial function?

The roots of a cubic polynomial can tell us the x-values where the polynomial function crosses the x-axis, or where the function has a y-value of zero. This can help us determine the behavior of the function, such as whether it is increasing or decreasing, and the number of turning points it has.

5. Are there any special cases for finding the roots of a cubic polynomial?

Yes, there are two special cases when finding the roots of a cubic polynomial. The first is when all three roots are real and distinct, which is the most common case. The second is when one root is real and the other two roots are complex conjugates, which means they are a pair of complex numbers with the same real part and opposite imaginary parts. In this case, the polynomial can be factored into a linear and quadratic factor with the real root and the complex conjugate roots, respectively.

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