Question

Use synthetic division to solve: `(x^2+7x-1)` divided by `(x+1)` ?

Answer 1

`(x^2+7x-1)/(x+1) = x+6-7/(x+1)`

Explanation:

We start by writing the coefficients of the dividend inside an L shape and the zero associated with the divisor just outside:

`-1color(white)(" ")"|"color(white)(" ")1color(white)(" ")7color(white)(" ")color(black)(-1)`
`color(white)(-1" ")"|"underline(color(white)(" "1" "7" "-1)`

Carry the first coefficient from the dividend down to below the line:

`-1color(white)(" ")"|"color(white)(" ")1color(white)(" ")7color(white)(" ")color(black)(-1)`
`color(white)(-1" ")"|"underline(color(white)(" "1" "7" "-1))`
`color(white)(-1" ")color(white)("|")color(white)(" ")1`

Multiply this first coefficient of the quotient by the test zero and write it in the second column:

`-1color(white)(" ")"|"color(white)(" ")1color(white)(" "-)7color(white)(" ")color(black)(-1)`
`color(white)(-1" ")"|"underline(color(white)(" "1" ")-1color(white)(" "-1))`
`color(white)(-1" ")color(white)("|")color(white)(" ")1`

Add up the second column and write the sum as the next term of the quotient:

`-1color(white)(" ")"|"color(white)(" ")1color(white)(" "-)7color(white)(" ")color(black)(-1)`
`color(white)(-1" ")"|"underline(color(white)(" "1" ")-1color(white)(" "-1))`
`color(white)(-1" ")color(white)("|")color(white)(" ")1color(white)(" "-)6`

Multiply this second coefficient of the quotient by the test zero and write it in the third column:

`-1color(white)(" ")"|"color(white)(" ")1color(white)(" "-)7color(white)(" ")color(black)(-1)`
`color(white)(-1" ")"|"underline(color(white)(" "1" ")-1color(white)(" ")color(black)(-6)`
`color(white)(-1" ")color(white)("|")color(white)(" ")1color(white)(" "-)6`

Add up the third column to give the remainder:

`-1color(white)(" ")"|"color(white)(" ")1color(white)(" "-)7color(white)(" ")color(black)(-1)`
`color(white)(-1" ")"|"underline(color(white)(" "1" ")-1color(white)(" ")color(black)(-6)`
`color(white)(-1" ")color(white)("|")color(white)(" ")1color(white)(" "-)6color(white)(" ")color(red)(-7)`

Reading off the coefficients, we have found:

`(x^2+7x-1)/(x+1) = x+6-7/(x+1)`