Question

Use Horner's rule to evaluate the following polynomial at x=3. Show your work. p(x)=x⁵ −2x⁴ +x³ −3x² +4x+1 (2) (a) Find the formula for the 4th degree Taylor polynomial, at x⁰ =0, for f(x)=eˣ

x
.

Answer 1

To evaluate the polynomial p(x)=x⁵−2x⁴+x³−3x²+4x+1 at x=3 using Horner's rule, you start by rewriting it in a different form and substitute x=3 into this expression, simplifying it step by step.

Step-by-step explanation:

Horner's rule is a method used to evaluate polynomials efficiently. To evaluate the polynomial p(x)=x⁵−2x⁴+x³−3x²+4x+1 at x=3 using Horner's rule, you start by rewriting it in a different form: p(x)=(x(x(x(x(x-2)+1)-3)+4)+1. Then, you substitute x=3 into this expression and simplify it step by step. Starting from the innermost parentheses, you replace the x variable with 3 and perform the calculations until you reach the outermost parentheses.

Using Horner's rule, the evaluation will be:
p(3)=(((3)(3)(3)(3-2)+1)-3)+4)+1
=(((81)(1)+1)-3)+4)+1
=(((81+1)-3)+4)+1
=((82-3)+4)+1
=(79+4)+1
=83+1
=84