Find the Taylor series generated by f at x = a.
-$$f ( x ) = x ^ { 3 } + 8 x ^ { 2 } + 7 x + 6 , a = - 2$$
A) $( x + 2 ) ^ { 3 } + 2 ( x + 2 ) ^ { 2 } - 13 ( x + 2 ) + 16$
B) $( x + 2 ) ^ { 3 } - 2 ( x + 2 ) ^ { 2 } - 13 ( x + 2 ) - 16$
C) $( x + 2 ) ^ { 3 } - 2 ( x + 2 ) ^ { 2 } - 27 ( x + 2 ) - 16$
D) $( x + 2 ) ^ { 3 } + 2 ( x + 2 ) ^ { 2 } - 27 ( x + 2 ) + 16$

Question

Quizplus · Accepted Answer

The Taylor series of a polynomial is the polynomial itself, but centered at the point $a$. Given $f(x) = x^3 + 8x^2 + 7x + 6$ and $a = -2$, we substitute $x = x + 2 - 2$ to center the polynomial at $a = -2$. Simplifying this directly gives the expression in option A.

Find the Taylor Series Generated by F at X = f(x)=x3+8x2+7x+6,a=−2f ( x ) = x ^ { 3 } + 8 x ^ { 2 } + 7 x + 6 , a = - 2f(x)=x3+8x2+7x+6,a=−2

Find the Taylor Series Generated by F at X = $f ( x ) = x ^ { 3 } + 8 x ^ { 2 } + 7 x + 6 , a = - 2$