Expand the polynomial P_5(x)=x^5+4x³+2 in powers of (x-1)

Expand the polynomial $P_5(x)=x^5+4x^3+2$ in powers of $(x-1)$.

Step 1: Derivatives at $x=1$

Since $P_5$ has degree 5, its Taylor expansion about $a=1$ is exact. Differentiating successively: \[P_5(1)=7,\quad P_5'(1)=17,\quad P_5''(1)=44,\quad P_5'''(1)=84,\quad P_5^{(4)}(1)=120,\quad P_5^{(5)}(1)=120. \tag{1}\]

Step 2: Form the Taylor coefficients

\[ \frac{7}{0!}=7,\ \ \frac{17}{1!}=17,\ \ \frac{44}{2!}=22,\ \ \frac{84}{3!}=14,\ \ \frac{120}{4!}=5,\ \ \frac{120}{5!}=1. \tag{2} \]

Step 3: Verification

\[x=1\text{: }7=1+4+2=7\ \checkmark\]

\[x=0\ (x-1=-1)\text{: }7-17+22-14+5-1=2=P_5(0)=0+0+2\ \checkmark\]