Question

How do you evaluate `f(x)=x+1/2x^3` at x=4 using direct substitution and synthetic division?

Answer 1

Direct Substitution : f(x) = f(4) = 36#

Synthetic Division :
Quotient `(1/2)x^2 + 2x + 9`, Remainder `= 18/(x-4)`

Explanation:

For x = 4,

`f(x) = f(4) = (1/2)x^3 + x = (1/2)4^4 + 4 = 36`

Using Synthetic Division,

`color(white)(aa)4color(white)(aa)|color(white)(aa)1/2color(white)(aa)0color(white)(aa)1color(white)(aa)0`
`color(white)(aaaaa)|color(white)(aa)darrcolor(white)(a)2color(white)(aa)8color(white)(aa)18`
`color(white)(aaaaaa)-------`
`color(white)(aaaaaaaaa)1/2color(white)(aa)2color(white)(aa)9color(white)(aa)18`

Quotient `(1/2)x^2 + 2x + 9`, Remainder `= 18/(x-4)`