Question

Expand ${\displaystyle 2x^3+7x^2+x-1}$ in powers of x - 2

Answer 1

Let  ${\displaystyle f\left(x\right)=2x^3+7x^2+x-1}$

Here a = 2

${\displaystyle f\left(x\right)=2x^3+7x^2+x-1}$	𝒇(𝟐)=𝟒𝟓
${\displaystyle f\prime \left(x\right)=6x^2+14x+1}$	𝒇′(𝟐)=𝟓𝟑
${\displaystyle f\prime \prime \left(x\right)=12x+14}$	𝒇′′(𝟐)=𝟑𝟖

𝒇′′′(𝒙)=𝒇′′′(𝟐)=𝟏𝟐

Taylor’s series is :

${\displaystyle f\left(x\right)=f\left(a\right)+(x-a)f\prime \left(a\right)+\frac{{(x-a)}^2}{2!}f\prime \prime \left(a\right)+....}$

${\displaystyle 2x^3+7x^2+x-1=45+(x-2)53+\frac{{(x-2)}^2}{2!}38+\frac{{(x-a)}^3}{3!}12}$

${\displaystyle 2x^3+7x^2+x-1=45+53(x-2)+19{(x-2)}^2+2{(x-2)}^3}$