Question

How do you use synthetic substitution to find f(-2) for `f(x)=x^4-4x^3-4x+6`?

Answer 1

`color(red)(f(-2) = 70)`

The Remainder Theorem states that when we divide a polynomial `f(x)` by `x-c` the remainder `R` equals `f(c)`.

We use synthetic division to divide `f(x)` by `x-c`, where `c = -2`.

Step 1. Write only the coefficients of `x` in the dividend inside an upside-down division symbol.

`|1" " -4" " "2" " " " "-4" " " " "6`
`|color(white)(1)`
`stackrel("————————————————————)`

Step 2. Put the divisor at the left.

`color(red)(-2)|1" " -4" " "2" " " " "-4" " " " "6`
`" " " |`
`" "stackrel("————————————————————)`

Step 3. Drop the first coefficient of the dividend below the division symbol.

`-2|1" " -4" " "2" " " " "-4" " " " "6`
`" " " |`
`" "stackrel("————————————————————)`
`" " " "color(red)(1)`

Step 4. Multiply the drop-down by the divisor, and put the result in the next column.

`-2|1" " -4" " " " "2"" "-4" "6`
`" " " |" " " " "color(red)(-2)`
`" " " "stackrel("————————————————————")`
`" " " "1`

Step 5. Add down the column.

`-3|1" " -4" " " " "2"" "-4" "6`
`" " " |" " " " "-2`
`" " " "stackrel("————————————————————)`
`" " " "1" " " "color(red)(-6)`

Step 6. Repeat Steps 4 and 5 until you can go no farther

`-2|1" " -4" " "2" " " " "-4" " " " "6`
`" " " |" " " " "-2" "12" "-28" " " " "64`
`" " " "stackrel("————————————————————)`
`" " " "1" "-6" " 14" "-32 " " " "color(red)(70)`

The remainder is `70`, so `f(-2) = 70`.

Check:

`x^4-4x^3 +2x^2 –4x +6 = (-2)^4-4(-2)^3+2(-2)^2-4(-2)+6 = 16+32 +8+8+6 = 70`