Question

How do you use synthetic substitution to find x=1 for `P(x)=4x^3-5x^2+3`?

Answer 1

`P(1)=2`. For a demonstration of how to show this, refer to the explanation below.

Explanation:

The remainder of the synthetic substitution will be equal to `P(1)`.

The terms in the top row of the division will be the coefficients of the terms. Don't forget the missing `x` term: `color(orange)(4)x^3color(orange)(-5)x^2+color(orange)0x+color(orange)3`

`{:(ul1"|",4,-5," 0"," 3"," "),(" ",ul" ",ul(" "color(red)4),ulcolor(blue)(-1),ulcolor(green)(-1),+),(" ",color(red)4,color(blue)(-1),color(green)(-1),"|"mathbf(" 2")," "):}`

The following is an explanation of the synthetic division:

Bring the original `4` term down. Then multiply it by `1` to get `4` again, which you bring up into the next column. Add the `4` to the `-5` up top to get `-1`, and repeat the pattern of multiplying by `1`, bringing to the next column and adding.

Since the remainder is `2`, we see that `P(1)=2`.

This also shows us that

`(4x^3-5x^2+3)/(x-1)=4x^2-x-1+2/(x-1)`