Question3. [15] a) Find the $5^{\text {th }}$ degree Taylor polynomial of $8 x^{5}-6 x^{4}+3 x^{3}+27 x^{2}+32 x-12$ about $x=0$ and [6] about $x=20$, then simplify each result. What do you observe? b) Find the $100^{\text {th }}$ degree Taylor polynomial of $8 x^{5}-6 x^{4}+3 x^{3}+27 x^{2}+32 x-12$ about $x=0$ [6] and about $x=20$, then simplify each result. What do you observe? c) What can you conclude about the $n^{\text {th }}$ degree Taylor polynomial about $x=a$ of a polynomial of [3] degree $m$, where $n \geq m$ ?

Studdy Solution
For a polynomial of degree $m$, the $n^{\text{th}}$ degree Taylor polynomial about any point $x = a$ where $n \geq m$ is the polynomial itself. This is because the Taylor polynomial of degree $n$ captures all terms up to degree $n$, and since $n \geq m$, it includes all terms of the original polynomial.
The observations for parts (a) and (b) show that the Taylor polynomial of a polynomial function, when the degree of the Taylor polynomial is greater than or equal to the degree of the polynomial, is the polynomial itself, regardless of the center point. This leads to the conclusion in part (c).