MHB Polynomials and Roots: Properties and Analysis

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The discussion focuses on the properties and analysis of polynomials and their roots. It introduces three polynomials: P(x), Q(x), and T(x), with roots that are shifted by -1 and +1 respectively. The sums of coefficients and alternate coefficients of these polynomials are calculated, revealing relationships between them based on the parity of their degree. An example polynomial is analyzed in detail, demonstrating how to derive new polynomials from existing ones by adjusting the roots. This method aids in understanding polynomial behavior and factorization.
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Hello.

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

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

Let \ r_1, r_2, \cdots, r_n, roots of the polynomial.

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

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

r_1-1, r_2-1, \cdots, r_n-1

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

r_1+1, r_2+1, \cdots, r_n+1

I will assume:

p_0=q_0=t_0=1Therefore:

\displaystyle\sum_{i=0}^n(p_i)=q_n

and

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

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

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

Example:

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

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

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

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

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

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.
 
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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:

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

Sum of coefficients=-995328

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

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

2º) Coefficient of x:

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

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

3º) Coefficient of x^2:

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

Sum of coefficients=-2853504.

4º) Coefficient of x^3:

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

Sum of coefficients=1177824.

5º) Coefficient of x^4:

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

Sum of coefficients=-149768.

6º) Coefficient of x^5:

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

Sum of coefficients=-17676.

7º) Coefficient of x^6:

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

Sum of coefficients=4634.

8º) Coefficient of x^7:

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

Sum of coefficients=-79.

9º) Coefficient of x^8:

\dfrac{P''''''''(x)}{8!}=9x-33

Sum of coefficients=-24.

10º) Coefficient of x^9:

\dfrac{P'''''''''(x)}{9!}=1.

Therefore, the resulting polynomial is:

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

Regards.
 
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