Concerning the polynomial p(x) = ao+a₁x + +anx", prove the following result. For a given x, we set (an, Pn, Yn) = (an, 0, 0) and define inductively (aj, B₁, y) = (a + xαj+1, α;+1+xBj+1, B₁+1 + xyj+1) for j=n-1, n-2,...,0. Then p(x) = ao, p'(x) = Bo, and p"(x) = 270.
Concerning the polynomial p(x) = ao+a₁x + +anx", prove the following result. For a given x, we set (an, Pn, Yn) = (an, 0, 0) and define inductively (aj, B₁, y) = (a + xαj+1, α;+1+xBj+1, B₁+1 + xyj+1) for j=n-1, n-2,...,0. Then p(x) = ao, p'(x) = Bo, and p"(x) = 270.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![16. Concerning the polynomial \( p(x) = a_0 + a_1 x + \cdots + a_n x^n \), prove the following result. For a given \( x \), we set \( (\alpha_n, \beta_n, \gamma_n) = (a_n, 0, 0) \) and define inductively
\[
(\alpha_j, \beta_j, \gamma_j) = (a_j + x \alpha_{j+1}, \alpha_{j+1} + x \beta_{j+1}, \beta_{j+1} + x \gamma_{j+1})
\]
for \( j = n-1, n-2, \ldots, 0 \). Then \( p(x) = \alpha_0 \), \( p'(x) = \beta_0 \), and \( p''(x) = 2\gamma_0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Fd17ae16c-0957-4688-9b7d-5654a8a2791b%2Frtf5wvn_processed.png&w=3840&q=75)
Transcribed Image Text:16. Concerning the polynomial \( p(x) = a_0 + a_1 x + \cdots + a_n x^n \), prove the following result. For a given \( x \), we set \( (\alpha_n, \beta_n, \gamma_n) = (a_n, 0, 0) \) and define inductively
\[
(\alpha_j, \beta_j, \gamma_j) = (a_j + x \alpha_{j+1}, \alpha_{j+1} + x \beta_{j+1}, \beta_{j+1} + x \gamma_{j+1})
\]
for \( j = n-1, n-2, \ldots, 0 \). Then \( p(x) = \alpha_0 \), \( p'(x) = \beta_0 \), and \( p''(x) = 2\gamma_0 \).
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