Ruffini's rule is a method for dividing a polynomial P(x) by a binomial x-a. The document provides two examples of using Ruffini's rule to divide polynomials. It involves writing the coefficients of the polynomials, then multiplying and subtracting terms of the divisor from the dividend according to the rule to obtain the quotient and remainder.

1. RUFFINI'S RULE
We use Ruffini's rule when we want to divide a
polinomial P(x) by a binomial x-a

2. Let's divide
( x5−3 x3+x2−x+5):( x−2)
First, write only the coefficients;
if one term is missing, the coefficient is 0

3. Let's divide
( x5−3 x3+x2−x+5):( x−2)
First, write only the coefficients;
if one term is missing, the coefficient is 0
1 0 -3 1 -1 5

4. Let's divide
( x5−3 x3+x2−x+5):( x−2)
Then, the number a; in this case, a=2
1 0 -3 1 -1 5
2

5. Let's divide
( x5−3 x3+x2−x+5):( x−2)
The first term of the quotient will be:
x5 : x=x We only write the coefficient: 1
1 0 -3 1 -1 5
2
1

6. Let's divide
( x5−3 x3+x2−x+5):( x−2)
The terms of the quotient are to be multiplied
by the divisor and subtracted to the dividend
1 0 -3 1 -1 5
2
1
2
2

7. Let's divide
( x5−3 x3+x2−x+5):( x−2)
Don't worry about the minus sign when subtracting
because you already changed it writing 2 instead of -2
1 0 -3 1 -1 5
2
1
2
2

8. Let's divide
( x5−3 x3+x2−x+5):( x−2)
You already have the second term of the quotient.
Go on the same way multiplying 2 · 2.
1 0 -3 1 -1 5
2
1
2
2

9. Let's divide
( x5−3 x3+x2−x+5):( x−2)
Go on the same way.
1 0 -3 1 -1 5
2
1
2
2
4
1

10. Let's divide
( x5−3 x3+x2−x+5):( x−2)
Go on the same way.
1 0 -3 1 -1 5
2
1
2
2
4
1
2
3

11. Let's divide
( x5−3 x3+x2−x+5):( x−2)
Go on the same way.
1 0 -3 1 -1 5
2
1
2
2
4
1
2
3
6
5

12. Let's divide
( x5−3 x3+x2−x+5):( x−2)
Go on the same way.
1 0 -3 1 -1 5
2
1
2
2
4
1
2
3
6
5
10
15

13. Let's divide
( x5−3 x3+x2−x+5):( x−2)
This last term is the remainder.
1 0 -3 1 -1 5
2
1
2
2
4
1
2
3
6
5
10
15

14. Let's divide
( x5−3 x3+x2−x+5):( x−2)
This last term is the remainder.
1 0 -3 1 -1 5
2
1
2
2
4
1
2
3
6
5
10
15=R

15. Let's divide
( x5−3 x3+x2−x+5):( x−2)
Write the quotient; the last term is the constant
The term before it is the x term and so on
1 0 -3 1 -1 5
2
1
2
2
4
1
2
3
6
5
10
15=R

16. Let's divide
( x5−3 x3+x2−x+5):( x−2)
Q( x)=x4+2 x3+x2+3x+5
1 0 -3 1 -1 5
2
1
2
2
4
1
2
3
6
5
10
15=R

17. ● Another example; in this case, the number a
will be negative, so that the divisor will be in the
form x+a, being a a negative number

18. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)

19. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
First, write only the coefficients;
if one term is missing, the coefficient is 0

20. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
First, write only the coefficients;
if one term is missing, the coefficient is 0
2 1 0 4 -3 1

21. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
Now write a: remember a=-1
2 1 0 4 -3 1

22. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
Now write a: remember a=-1
2 1 0 4 -3 1
-1

23. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And proceed to the division
2 1 0 4 -3 1
-1

24. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And proceed to the division
2 1 0 4 -3 1
-1
2

25. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And proceed to the division
2 1 0 4 -3 1
-1
2
-2
-1

26. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And proceed to the division
2 1 0 4 -3 1
-1
2
-2
-1
1
1

27. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And proceed to the division
2 1 0 4 -3 1
-1
2
-2
-1
1
1
-1
3

28. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And proceed to the division
2 1 0 4 -3 1
-1
2
-2
-1
1
1
-1
3
-3
-6

29. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And proceed to the division
2 1 0 4 -3 1
-1
2
-2
-1
1
1
-1
3
-3
-6
6
7

30. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
Write the remainder
2 1 0 4 -3 1
-1
2
-2
-1
1
1
-1
3
-3
-6
6
7

31. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
Write the remainder
2 1 0 4 -3 1
-1
2
-2
-1
1
1
-1
3
-3
-6
6
7=R

32. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And the quotient:
2 1 0 4 -3 1
-1
2
-2
-1
1
1
-1
3
-3
-6
6
7=R

33. Let's divide
(2x5+ x4+4 x2−3x+1):( x+1)
And the quotient:
Q( x)=2 x4−x3+x2+3 x−6
2 1 0 4 -3 1
-1
2
-2
-1
1
1
-1
3
-3
-6
6
7=R