Question

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

Answer 1

The remainder is `-31` and the quotient is `=-4x^2-8x-13`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `2` `|` `color(white)(aaaa)` `-4` `color(white)(aaaaa)` `0` `color(white)(aaaaaa)` `3` `color(white)(aaaaaaa)` `-5`

`color(white)(aaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaaa)` `-8` `color(white)(aaaa)` `-16` `color(white)(aaaaa)` `-26`

`color(white)(aaaaaaaaa)`##_________

`color(white)(aaaaa)` `|` `color(white)(aaaa)` `-4` `color(white)(aaaa)` `-8` `color(white)(aaaa)` `-13` `color(white)(aaaaa)` `color(red)(-31)`

The remainder is `-31` and the quotient is `=-4x^2-8x-13`

`(-4x^3+3x-5)/(x-2)=-4x^2-8x-13-31/(x-2)`

Apply the remainder theorem

When a polynomial `f(x)` is divided by `(x-c)`, we get

`f(x)=(x-c)q(x)+r`

Let `x=c`

Then,

`f(c)=0+r`

Here,

`f(x)=-4x^3+3x-5`

Therefore,

`f(2)=-4*2^3+3*2-5`

`=-32+6-5`

`=-31`