This document provides instruction on dividing polynomials using long division and synthetic division methods. It begins with objectives and instructions, then provides examples of dividing polynomials using standard long division and synthetic division. Examples are worked through step-by-step. Synthetic division is described as a shortcut method that uses only the numerical coefficients of terms, while long division follows the process of polynomial division. The document compares and contrasts the two methods, noting that synthetic division is easier but requires finalizing partial quotients. Students are provided homework to practice both division methods.

1. ENHANCED
MATHEMATICS
TEACHING AND
LEARNING PROGRAM
(EMTAP)

2. For today, you will learn about:
Terminal Objective:
division of polynomials using
long division and synthetic
division.

3. At the end of the session, you must be able to:
Enabling Objective:
a. Arrange polynomials in standard form
and identify its numerical coefficients;
c. Compare and contrast standard long
division of polynomials method with
synthetic division technique.
b. Divide polynomial by a binomial using
long and synthetic division methods; and

4. 1. First, is to find your comfortable
place with strong internet connection
and free from disturbances.
Remember:
2. Second, have your modules,
learning materials and other resources
ready.
3. Finally, mute your device to avoid
interrupting the presentation. Unmute
only when asked to do so.

5. TEACHER III
FERMELDY NATIONAL HIGH SCHOOL
Standard Long and
Synthetic Division of
Polynomials

6. Starting Activity 1: Fundamentals with Monomials
____1) x – x
Warm-up
Direction: Perform the indicated operation for each
expression. Recall the rules in the fundamental
operations of monomials.
____2) x • x
____3) x ÷ x
____4) 5x2 – (-2x2)
____5) -4x3 – 3x3
____6) -4x3 – (-3x3)
____7) x3 • x
____8) 24x3 ÷ 4x
____9) 5x3 • 3x2
____10) 16x5 ÷ (-4x2)
0
x2
1
7x2
-7x3
-x3
x4
6x2
15x5
-4x3

7. Starting Activity 2: Set Your Standards
1. 3x4 – 5x10 + 2x2 – 6x – 3
Warm-up
Direction: Write the standard form (SF) of the following
polynomials. After which, identify in order the numerical
coefficients (NC) in each term including the constant.
2. 5x4 – 3x6 + 7
3. - 4x + 7x8 – 3x4
4. x4 + 2x3 + 7x – 5x2 + 3
SF: – 5x10 + 3x4 + 2x2 – 6x – 3
NC: – 5, 3, 2 – 6 – 3
SF: – 3x6 + 5x4 + 7
NC: -3, 5, 7
SF: 7x8 – 3x4 - 4x
NC: 7, -3, -4
SF: x4 + 2x3 – 5x2 + 7x + 3
NC: 1, 2, -5, 7, 3

8. Dividing polynomials is much like
the familiar process of dividing
numbers.
When we divide 38 by 7, the quotient
is 5 and the remainder is 3.
Let us start here!
We write:

9. Standard Long
Division
Method 1

10. Standard Long Division
If P(x) and D(x) are polynomials, with D(x) ≠ 0, then
there exist unique polynomials Q(x) and R(x), where
R(x) is either 0 or of degree less than the degree of
D(x), such that:
or P(x) = D(x) . Q(x) + R(x)
The polynomials P(x) and D(x) are called the dividend
and divisor, respectively.
Q(x) is the quotient and R(x) is the remainder.

11. Example 1
Divide x4 + 3x3 -4x2 –x +1 by x + 2
8. Write the remainder over divisor

12. The division process ends when
the last line is of lesser degree than
the divisor. The last line then
contains the remainder. The top
line contains the quotient.
Note:

13. Example 2
2. Divide 2x2 + x3 +1 by x – 1
The terms
made into
standard form
Added 0x as the
third term since
there is no term for
x or linear term.
8. Write the remainder over divisor

14. Try this on your own! Example 3
3. Divide 6x2 – 26x + 12 by x – 4
v
8. Write the remainder over divisor

15. 1. What difficulty did you encounter
in long division of polynomials?
Checkpoint!
2. What did you do to overcome this
difficulty?

16. Synthetic
Division
Method 2

17. Synthetic Division
Synthetic division is a shortcut method for
polynomial division which can be used in
place of the standard long division.
It uses only the numerical coefficients of
the terms of the dividend and the divisor,
and multiplication and addition as the
means of operation.

18. Example 1
Divide x4 – 3x3 + x2 – 2x + 3) ÷ (x – 2)
v
v
v

19. Example 1
Divide x4 – 3x3 + x2 – 2x + 3) ÷ (x – 2)
v
v
v
v
v
v
v
v
9
10. Write the remainder over divisor
-4
Note: The results in step 8 (1, -1,
-1, -4, -5) are the numerical
coefficients of the terms of the
quotient in descending order.
Since the divisor is linear, the
degree of the quotient is one
degree less than the dividend.
The first term is the leading
coefficient of the dividend and
the last term is the remainder

20. Try this on your own. Example 2
Divide 2x2 + x3 + 1 by x + 1
1
v
-1
v
1
v
-1

21. Try this on your own. Example 2
Divide 2x2 + x3 + 1 by x + 1
1
v
-1
v
1
v
-1
v
-1
v
1
v
2
9
1x2 + 1x – 1 + or simply
x2 + x – 1 +
10. Write the remainder over divisor

22. 1. Compare and contrast long division and
synthetic division
Let’s Compare
2. Which method is easier, long division or
synthetic division? Why?

23. Directions: Below is an illustration of long division and synthetic
division wherein the divisor is in the form ax – c (x = 𝑐/𝑎). Study
the steps well then answer the questions that follow.
Let’s Dig Deeper

24. Let’s Dig Deeper
1. How do you compare the quotients in both methods?
Do you think they are equal?
2. What will you do with the numerical coefficients of the terms of the
quotient in synthetic division so that it would be the same with that of long
division?
3. How do you compare the number you used to divide the numerical
coefficients of the terms of the quotient in synthetic division with that of
the numerical coefficient of x in the divisor?
4.What can you say about the remainders in both methods?

25. Note:
There are cases when the quotient in long
division differs with the quotient in synthetic
division. This happens whenever the numerical
coefficient of the leading term of the divisor is
not equal to 1.
“Partial quotient” in synthetics division means
that it’s tentative or not the final quotient yet.
In short, if the divisor is in the form ax – c
where a ≠ 1, divide the quotient by “a” and
leave the remainder as it is.

26. When dividing polynomials…
To sum-up:
Synthetic division is a shortcut method for
polynomial division which can be used in place of
the standard long division. It uses only the
numerical coefficients of the terms of the dividend
and the divisor, and multiplication and addition as
the means of operation.
Long division of polynomials can be obtained by
or P(x) = D(x) . Q(x) + R(x).
You are free to choose the method by which they
can find comfort in solving.
Whenever you are using synthetic division, make
sure to finalize the partial quotient.

27. Take this home
Divide the following polynomials using
the two methods.
1. (x3– x2 + x – 1) ÷ (x + 2)
2. (– 3x3 + 2x4 + x – 1) ÷ (x + 2)

28. Thank
You!