notes / Integrated Mathematics / chapter 10: Polynomials
10.5 Synthetic Division
When the divisor is , long division can be cut down to arithmetic on the coefficients alone.
(Also known as Ruffini’s rule.)
Consider , where and . Then synthetic division is given by
Note this is , not “”.
Consider divided by :
Each shape denotes a row.
* This only works when the divisor is , i.e. its leading coefficient is .
Filling it in: bring down the ; multiply by and write the product under the next coefficient; add down the column; repeat.
the long division from before: quotient , remainder .
Problem 10.5.1. Divide by .
Show solution
By long division:
By synthetic division (note ):
Problem 10.5.2. Divide into .
Show solution
So
(dividing by ). So we have , .
Let’s test!
multiplying by , we get
So : dividing by gives quotient and remainder .
Problem 10.5.3. Use synthetic division:
(a) ;
(b) .
Show solution
(a)
(b) , so divide by and then by :
This gives quotient and remainder . Check:
Exercises
Exercise 10.5.4. Divide by .
Show solution
Exercise 10.5.5. Divide by .
Show solution
Exercise 10.5.6. Divide by .
Show solution
:
Dividing the quotient row by :
Exercise 10.5.7. Divide by .
Show solution
(, so .) Dividing the quotient row by :
Exercise 10.5.8. Divide by .
Show solution
, so divide by and negate the quotient:
Exercise 10.5.9. (The statement isn’t copied in the notes: it shows a synthetic division of by that went wrong.)
Show solution
Write :
He forgot to turn into .