Question

How do you use synthetic division to divide `( 6x^3 - x^2 - 7x + 5 ) / ( x - 3 )`?

Answer 1

`(6x^3-x^2-7x+5)/(x-3) = color(blue)(6x^2+17x+44)+color(red)(137/(x-3))`

Explanation:

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

`color(white)(1)|6" "-1color(white)(X)-7" "" "5`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`

Step 2. Put the divisor at the left.

`3|6" "-1color(white)(X)-7" "" "5`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`

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

`3|6" "-1color(white)(X)-7" "" "5`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`
`color(white)(1)|color(red)6`

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

`3|6" "-1color(white)(X)-7" "" "5`
`color(white)(1)|" "color(white)(X1)18`
`" "stackrel("—————————————)`
`color(white)(1)|color(blue)6`

Step 5. Add down the column.

`3|6" "-1color(white)(X)-7" "" "5`
`color(white)(1)|" "color(white)(X1)18" "color(white)(1)51color(white)(X1)132`
`" "stackrel("—————————————)`
`color(white)(1)|color(blue)6" "color(red)(color(white)(X)17`

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

`3|6" "-1color(white)(X)-7" "" "5`
`color(white)(1)|" "color(white)(X1)18" "color(white)(1)51color(white)(X1)132`
`" "stackrel("—————————————)`
`color(white)(1)|color(blue)(6)" "color(white)(X)17" "color(white)(1)44" "color(white)(1)color(red)(137)`

Answer:

`(6x^3-x^2-7x+5)/(x-3) = 6x^2+17x+44+ (137)/(x-3)`

Check:

`(x-3)(6x^2+17x+44+137/(x-3)) = (x-3)(6x^2+17x+44)+137 = 6x^3+17x^2+44x-18x^2-51x-132+137= 6x^3-x^2-7x+5`

It works!