Chapter 8, Problem 1RE

Describe how Horner’s Algorithm evaluates f(x) when

1. $f\left(x\right)=2x^3-4x+1$ and $x=-3$

2. $f\left(x\right)=x^4-2x^3+x^2-5x+6$ and $x=2$

To determine

Horner’s Algorithm evaluates f(x) when,

$f(x)=x^4-2x^3+x^2-5x+6$ and $x=-3$.

Explanation of Solution

Horner’s evaluated the algorithm $f(x)$ for each parts.

Horner’s Algorithm evaluates the expression $a_0+a_{1}x+a_2{x}^2+\mathrm{......}+a_n{x}^n$,

Given that,

Integers $2,0+2(-3)=-6,-4+(-6)(-3)=14,1+14(-3)=-41=f(-3)$,……, $a_n$ and an integer $n$, by:

1. Set  $S=a_n$

2. For, $i$ from 1 to $n$, modify the value of $S$ to the value $a_{n-1}+sx$.

3. Output the value of $S$.

Given that,

$f(x)=2x^3-4x+1$

And

$x=-3$

Horner’s Algorithm will successively,

Set to, $S$

$2,0+2(-3)=-6,-4+(-6)(-3)=14,1+14(-3)=-41=f(-3)$

To determine

Horner’s Algorithm evaluates f(x) when,

$f(x)=x^4-2x^3+x^2-5x+6$ and

$x=2$

Explanation of Solution

Given that,

$f(x)=x^4-2x^3+x^2-5x+6$

And

$x=2$

Horner’s Algorithm will successively, set to, $S$

$1,-2+1(2)=0,1+0(2)=1,-5+1(2)=-3,6-3(2)=0=f(2)$