Answered: Show that every polynomial of degree n:…

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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Show that every polynomial of degree n:
y = f(x) = c_nxⁿ + c_n-1x^n-1 + . . . + c₂x² + c₁x + c₀
is a function by mathematical induction on degree n.
Assume n is a nonnegative integer, all c_is are real, c_n ≠ 0, and x and y are also real.

Some hints: use the definition: f is a function iff a = b implies f(a) = f(b) and recall that in informal proofs we show an implication by assuming the if part of the implication, and then deducing the then part of the implication.

The base case will show that a = b implies f(a) = f(b) when f(x) = c₀ (a constant function). The inductive case will assume a = b implies f(a) = f(b) for degree k, and will deduce it is also true for degree k+1.

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