10

Mathematics
Quarter 1 – Module 5
Polynomials

1
Mathematics – Grade 10
Alternative Delivery Mode
Quarter 1 – Module 5: Polynomials

First Edition, 2021

Republic Act 8293, Section 176 states that: No copyright shall subsist in any work of the
Government of the Philippines. However, prior approval of the government agency or office
wherein the work is created shall be necessary for exploitation of such work for profit. Such
agency or office may, among other things, impose as a condition the payment of royalties.

Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.)
included in this book are owned by their respective copyright holders. Every effort has been
exerted to locate and seek permission to use these materials from their respective copyright
owners. The publisher and authors do not represent nor claim ownership over them.

Published by the Department of Education

Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writers: Ruby Ann H. Licot MAEM/MAT-Math

Joan C. Reyes MAED

Editors: Marcela R. Bautista Dev.EdD

Ramil A. Escaso EdD
Filipina A. Trazo
Management Team: Bianito A. Dagatan EdD, CESO V
Schools Division Superintendent

Faustino N. Toradio PhD

Assistant Schools Division Superintendent

Felix C. Galacio Jr. PhD

EPS, Mathematics

Josephine D. Eronico PhD

EPS, LRMDS

Printed in the Philippines by Schools Division of Bohol

Department of Education – Region VII, Central Visayas

Office Address: 0050 Lino Chatto Drive Barangay Cogon, Tagbilaran City,
Bohol
Telefax: (038) 501 – 7550
Tel Nos. (038) 412 – 4938; (038) 411-2544; (038) 501 – 7550
E-mail Address: depedbohol@deped.gov.ph

2
 Learning Competencies: Perform division of polynomials using long division and synthetic
division. (M10AL-Ig-1)

At the end of the lesson, you are expected to:

 determine the remainder of the polynomial equation by long division method.
 find the remainder of polynomials using synthetic division method.
 use the Remainder Theorem to find the remainder when polynomial is divided by (x - c).
 determine whether (x – c) is a factor of the polynomial P(x) by using the Factor Theorem.

Lesson 1: Long Division

What is it

Consider the examples below:

1.) Long Division of Integers 2.) Long Division of P(x) ÷ x-c

107 x2 - x + 2
5 ) 538 x – 1) x – 2x2 + 3x +2
3

5 +
3 x3 (–) x2
0 - x2 + 3x
38 + -
35 (-) x2 (+) x
3 remainder 2x + 2
- +
The quotient here is 107 and the remainder is 3. (+)2x (-) 2

Remainder
4
The quotient here is x2 – x + 2 and the remainder is 4.

To check whether the answer is right or wrong, use the Division Algorithm for Polynomials.
P(x) = (x-c).Q(x) + R
In example number 2, (x-c) is x-1 which is the divisor, Q(x) is x2 – x +2 which is the quotient and
R is 4, the remainder.

Check by using the division algorithm, that is:

P(x) = (x-c).Q(x) + R
P(x) = (x-1) (x2 – x +2) +4
P(x) = x3 – x2 +2x –x2 +x - 2 +4
P(x) = x3 – 2x2 +3x +2

What’s More

Direction: Use the Long Division Method to find the remainder of the following polynomials.

1.) (x3 - 4x2 + x +6) ÷ (x-2) 2.) (2x2 + 9x-3) ÷ (x+3)

3
 Lesson 2: Synthetic Division

What is it
Synthetic Division is a sequence of substitution operations used to evaluate a polynomial
function of any degree.

In solving for the value of the polynomial function by using synthetic division, simply
follow the steps below given P(x) = 2x3 -8x2 + 19x -12 is divided by x – 3

1) Solve the value of x in the divisor and put it in the window. This will serve as the
multiplier of the coefficients of the variable of the polynomial P(x)
The divisor is x – 3. To solve for x, equate the linear equation to zero then, solve for x.
x–3=0
x=3
2) Arrange the polynomial base on the exponents in descending order. Get the
coefficients of the variable in polynomial P(x).
In P(x) = 2x3 -8x2 + 19x -12, the exponents are already arranged in descending
order. Get the coefficients, and place it below the multiplier that is:
2 -8 19 -12
3

3) Put a space after the coefficients. Then, put a line and start the multiplication. First,
bring down the first coefficient, then multiply it to the multiplier and put the answer
on the space in line with the next coefficient then add the two numbers. Continue with
the same process. The last sum is the remainder.
2 -8 19 -12
3 6 -6 39
2 -2 13 27 Remainder

 If in case, the next degree of the function is cannot be seen. Write zero as the
coefficient of the variable.

4) The remainder R obtained in synthetic division of P(x) by (x – c), provides this

information:
a) The remainder R gives the value of P(x) at x-c, that is, R = P(c).
b) If R = 0, then x- c is a factor of P(x).
c) If R = 0, then (c, 0) is an x intercept of the graph of P(x).
d) This means that the value of P (3) = 27

Example:

1)
If the polynomial P(x) = x3 + 9x -8x2-1 is divided by x + 2
Solution:
First, arrange the polynomial function base on the exponents.

-2 1 -8 9 -1

-2 20 -58

1 -10 29 -59 Remainder R =-59

The remainder is -59. Therefore, P (-2) = -59.

4
2) The polynomial P (a) = 3a3 + 9a -15 at a = - 4.

-4 3 0 9 -15

-12 48 -228

3 -12 57 -243 Remainder R= -243

The remainder is -35, and therefore P(-4) = -243.

What’s More
Find the remainder of the polynomial function using synthetic division.

1) P(x) = x3 – 2x2 + 3x +2 ;x=1

2) P(x) = x3 – 2x2 -75 ;x=5
3) P(x) = 2x2 + 9x-3 ; x = -3
4) P(x) = x3 + 8x2 +13x -10 ; x = -5

Lesson 3: Remainder Theorem and Factor Theorem

What is it

Remainder Theorem- if a polynomial P(x) is divided by x - c where c is a real number, then

the remainder is P(c).
Proof:
P(x) = ( x - c). Q(x) + R Division Algorithm for Polynomials
P(c) = (c-c). Q(c) + R The equation is true for all x, therefore, let x=c.
P(c) = 0. Q(c) + R
P(c) = R Therefore, the remainder R is equal to P(c).

Examples:

1. Apply the Remainder Theorem to find the remainder of (x3- 3x2+x+4) ÷ (x-2)

Solutions: Check by using the Long Division Method

P(x) = x3- 3x2+x+4 x2 – x - 1
Since x- c = x-2 x-2) x3- 3x2 + x + 4
Then, c = 2 or x= 2 x3 - 2x2
Therefore, -x2 + x
P(2)= (2)3 – 3(2)2 +(2) +4 -x2 +2x
= 8- 12 +2 +4 -x+4
= 2 remainder -x+2
2 remainder

5
2. Use the Remainder Theorem to find the remainder when x4 -3x3 +3x – 4 is divided by x+3
Solution:
P (x) = x4 -3x3 +3x – 4
c=-3
P (-3) = (-3)4 – 3(-3)3 + 3(-3) – 4
= 81 +81 – 9 – 4
= 141 remainder

The factor Theorem states that in the polynomial P(x), if P(c) = 0, where c is a real number, then
(x – c) is a factor of P(x). This means that if the remainder of the polynomial P(x) divided by (x – c) is
zero, then (x – c) is a factor of P(x).

1. When P(x) = x3 –x2 - 4x +4 is divided by x -2, the remainder is 0; that is, P(2) =0, then x –
2 is a factor to P(x) = x3 –x2 - 4x +4.

By using synthetic division, the same result is obtained.

2 1 -1 -4 4

2 2 -4

1 1 -2 0 Remainder

The remainder is 0, therefore x-2 is a factor of P(x) = x3 - x2 -4x +4.

2. In this example we can use the Remainder Theorem to determine whether it is a factor or
not given P(x) = x3 + x2 +x +2 divided by x+1.

P(x) = x3 + x2 +x +2

P(-1) = (-1)3 + (-1)2 +(-1) +2

P(-1) = -1 + 1 -1 +2

P(-1) = 1

The remainder is 1, therefore x+1 is not a factor of P(x) = x3 + x2 +x +2.

What’s More
A. Find the remainder of the following polynomials by using the Remainder Theorem.

1. (x3 + 14x2 + 47x – 12) ÷ (x+7)

2. (3y3 + 2y2 – y +5) ÷ (y – 5)

B. Use the factor theorem to determine whether the first polynomial is a factor of the second.

1. x– 2; 4x3 – 3x2 - 8x +4
2. x + 3; 2x3 + x2 – 13x +6

6
 Assessment

Directions: Read and understand the problems carefully. Write your answer on the answer sheet provided
for you. STRICTLY NO ERASURE.

1. What is the quotient when x2-25 is divided by x-5?

A. x-5 B. x-25 C. x + 5 D. x + 25

2. What is the remainder when P(x) = 4x2 + 2x -5 is divided by x – 2?

A. 25 B. 20 C. 15 D. 10

3. The quotient when (2x4 + 4x3 -5x2 + 2x -3) divided by (2x2 + 1)?
A. x2 + 2x -3 B. x2 - 2x + 3 C. x2 - 2x -3 D. x2 + 2x + 3

4. Which of the following gives a quotient of (3x2 + 2x + 4) and remainder of 19 when

divided by 2x-3?
A. 6x3 -5x2 +2x C. 6x3 -5x2 + 2x +7
B. B. 6x3 -5x2 +4x +7 D. 6x3 + 5x2 +2x + 7

Use illustration below to answer numbers 5-8.

To find the value of the polynomial P(x) = x3 + 4x2 + x + 4 for x = -4
The following computation is shown
1 4 1 4
-4
-4 0 -4
1 0 1 0
5. What is the value of the polynomial?
A. 4 B. -4 C. 1 D. 0

6.. In the computation shown above in number 5, what does the third row 1 0 1 0 stands for?
3
A. x + 1 B. the quotient C. the remainder D. the divisor

7. What is the quotient?

A. x3 + 1 B. x2 +1 C. x+ 4 D. x- 4

8. What is the divisor?

A. x3 + 1 B. x2 +1 C. x+ 4 D. x- 4

9. What is the quotient when (x2 -14x +49) is divided by (x-7)?

A. x + 7 B. x -49 C. x-7 D. x+ 40

10. Which of the following polynomials is not divisible by (x-1)?

A. 2x2 -5x +7 B. 2x2 +9x +1 C. 4x2 -16 D. all of the above

Reference

Mathematics Grade 10 Learner’s Module, Author: Melvin M. Callanta, et. al.

7
 Answer Sheet
Quarter 1-Module 5
Name: ___________________________________________ Score: ______________
Grade & Section: __________________________________

Lesson 1
What’s More
1.
2.

Lesson 2
What’s More
A.1.
2.
B.
1.
2.

Lesson 3
What’s More
A. 1.
2.
B. 1.
2.

Assessment
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

8
9
1