Question

How do you use synthetic division to divide `4x^3 + 7x^2 – 13x – 3` by `x-7`?

Answer 1

`(4x^3+7x^2-13x-3)/(x-7) = color(blue)(4x^2+35x+232)+color(red)(1621)/(x-7)`

Explanation:

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

`|4" "" "7color(white)(1)-13" "" "-3`
`|color(white)(1)`
`stackrel("—————————————————)`

Step 2. Put the divisor at the left.

`color(red)(7)|4" "" "7color(white)(1)-13" "" "-3`
`color(white)(1)|color(white)(1)`
`" "stackrel("—————————————————)`

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

`7|4" "" "7color(white)(1)-13" "" "-3`
`color(white)(1)|color(white)(1)`
`" "stackrel("—————————————————)`
`color(white)(1)|color(red)(4)`

Step 4. Multiply the result by the constant, and put the product in the next column.

`7|4" "" "7color(white)(1)-13" "" "-3`
`color(white)(1)|" "" "color(red)(28)`
`" "stackrel("—————————————————)`
`color(white)(1)|4`

Step 5. Add down the column.

`7|4" "" "7color(white)(1)-13" "" "-3`
`color(white)(1)|" "" "28`
`" "stackrel("—————————————————)`
`color(white)(1)|4" "color(white)(1)color(red)(35)`

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

`7|4" "" "7color(white)(1)-13" "" "-3`
`color(white)(1)|" "" "28" "color(white)(1)245" "color(white)(1)1624`
`" "stackrel("—————————————————)`
`color(blue)(color(white)(1)|4" "color(white)(1)35" "color(white)(1)232)" "color(white)(1)color(red)(1621)`

`(4x^3+7x^2-13x-3)/(x-7) = 4x^2+35x+232+1621/(x-7)`

Check:

`(x-7)(4x^2+35x+232+1621/(x-7)) = (x-7)(4x^2+35x+232)-1621 = 4x^3+35x^2+232x-28x^2-245x-1624+1621`

`= 4x^3+7x^2-13x-3`