Question

How do you evaluate `f(x)=5x^3+4x^2+8x+1` at x=2 using direct substitution and synthetic division?

Answer 1

`f(x) = f(2) = color(red)(73)`

Synthetic Division :
Quotient `color(brown)(Q = 5x^2 + 14x + 36)`, Remainder `color(red)(R = 73 / (x-2)`

Explanation:

Using direct substitution:

`f(x) =color(red)( f(2)) = 5*(2^3) + 4 * (2^2) + (8 * 2) + 1 = 40 + 16 + 16 + 1 = color(red)(73)`

Synthetic Division :

`color(white)(aa)2color(white)(aa)|color(white)(aaa)5color(white)(aaa)4color(white)(aaa)8color(white)(aaa)1`
`color(white)(aaaaaa)|color(white)(a)darrcolor(white)(aa)10color(white)(aa)28color(white)(aa)72`
`color(white)(aaaaaa)--------`
`color(white)(aaaaaaaaaa)5color(white)(aa)14color(white)(aa)36color(white)(aa)73`

Quotient `color(brown)(Q = 5x^2 + 14x + 36)`, Remainder `color(red)(R = 73 / (x-2)`