Polynomials and Roots: Properties and Analysis

[mente oscura]

Hello.

I open this 'thread', in number theory, but he also wears "calculation".

I've done a little research, I share with you.

[tex]Let \ r_1, r_2, \cdots, r_n[/tex], roots of the polynomial.

[tex]P(x)=p_0x^n+p_1x^{n-1}+ \cdots+p_{n-1}x+p_n[/tex]

[tex]Let \ Q(x)=q_0 x^n+q_1x^{n-1}+ \cdots +q_n[/tex], such that its roots are:

[tex]r_1-1, r_2-1, \cdots, r_n-1[/tex]

[tex]Let \ T(x)=t_0^n+t_1x^{n-1}+ \cdots +t_n[/tex], such that its roots are:

[tex]r_1+1, r_2+1, \cdots, r_n+1[/tex]

I will assume:

[tex]p_0=q_0=t_0=1[/tex]Therefore:

[tex]\displaystyle\sum_{i=0}^n(p_i)=q_n[/tex]

and

[tex]\displaystyle\sum_{i=0}^n(p_i)(-1)^{i+1}=t_n[/tex], if “n” it's even.

[tex]\displaystyle\sum_{i=0}^n(p_i)(-1)^i=t_n[/tex], if “n” it's odd.

Also I have found how to calculate "complete", recurrently cited polynomials.

Example:

[tex]P(x)=x^9-33x^8+149x^7+4431x^6-45669x^5+9081x^4+1506119x^3-7038363x^2+12556936x-7987980[/tex]

Sum of coefficients=-995328, addition and subtraction of alternate coefficients=-29030400

Roots:-11, -7, 2, 2, 3, 5, 7, 13, 19.[tex]Q(x)=x^9-24x^8-79x^7+4634x^6-17676x^5-149768x^4+1177824x^3-2853504x^2+2833920x-995328[/tex]

Sum of coefficients=0, addition and subtraction of alternate coefficients=-7987980

Roots:-12, -8, 1, 1, 2, 4, 6, 12, 18.[tex]T(x)=x^9-42x^8+449x^7+2380x^6-67152x^5+296240x^4+931632x^3-10983168x^2+30862080x-29030400[/tex]

Sum of coefficients=-7987980, addition and subtraction of alternate coefficients=-71442000

Roots:-10, -6, 3, 3, 4, 6, 8, 14, 20.

This procedure can be useful for the analysis of the possible roots of the polynomial, and its factorization.

Regards.

 

[mente oscura]

Hello.

Continuing with the topic.

I'm going to show, with an example, as it would generate a polynomial, with the roots of the original polynomial, decreased in 1 unit:

[tex]P(x)=x^9-33x^8+149x^7+4431x^6-45669x^5+9081x^4+1506119x^3-7038363x^2+12556936x-7987980[/tex]

Sum of coefficients=-995328

Roots:-11, -7, 2, 2, 3, 5, 7, 13, 19.[tex]Q(x)=x^9-24x^8-79x^7+4634x^6-17676x^5-149768x^4+1177824x^3-2853504x^2+2833920x-995328[/tex]

Roots:-12, -8, 1, 1, 2, 4, 6, 12, 18.
1º) Separate term=Sum of coefficients of P(x)=-995328

2º) Coefficient of [tex]x[/tex]:

[tex]\dfrac{P'(x)}{1}=9x^8-264x^7+1043x^6+26586x^5-228345x^4+36324x^3+4518357x^2-14076726x+12556936[/tex]

Coefficient of [tex]x[/tex], the new polynomialQ(x)=Sum of coefficients [tex]\dfrac{P'(x)}{1}[/tex]=2833920.

3º) Coefficient of [tex]x^2[/tex]:

[tex]\dfrac{P''(x)}{2}=36x^7-924x^6+3129x^5+66465x^4-456690x^3+54486x^2+4518357x-7038363[/tex].

Sum of coefficients=-2853504.

4º) Coefficient of [tex]x^3[/tex]:

[tex]\dfrac{P'''(x)}{3!}=84x^6-1848x^5+5215x^4+88620x^3-456690x^2+36324x+1506119[/tex].

Sum of coefficients=1177824.

5º) Coefficient of [tex]x^4[/tex]:

[tex]\dfrac{P''''(x)}{4!}=126x^5-2310x^4+5215x^3+66465x^2-228345x+9081[/tex].

Sum of coefficients=-149768.

6º) Coefficient of [tex]x^5[/tex]:

[tex]\dfrac{P'''''(x)}{5!}=126x^4-1848x^3+3129x^2+26586x-45669[/tex].

Sum of coefficients=-17676.

7º) Coefficient of [tex]x^6[/tex]:

[tex]\dfrac{P''''''(x)}{6!}=84x^3-924x^2+1043x+4431[/tex].

Sum of coefficients=4634.

8º) Coefficient of [tex]x^7[/tex]:

[tex]\dfrac{P'''''''(x)}{7!}=36x^2-264x+149[/tex]

Sum of coefficients=-79.

9º) Coefficient of [tex]x^8[/tex]:

[tex]\dfrac{P''''''''(x)}{8!}=9x-33[/tex]

Sum of coefficients=-24.

10º) Coefficient of [tex]x^9[/tex]:

[tex]\dfrac{P'''''''''(x)}{9!}=1[/tex].

Therefore, the resulting polynomial is:

[tex]Q(x)=x^9-24x^8-79x^7+4634x^6-17676x^5-149768x^4+1177824x^3-2853504x^2+2833920x-995328[/tex]

Regards.