Grade 8 Book 1 Unit 1 and 2

7. Division of Polynomial

If we are dividing a polynomial by another polynomial, it does not help to
rewrite each term over the denominator or the divisor. As an alternative, use the
long division in the way it is used for division of integers.

numerator dividend

+ + + + +

+ divisor

denominator

Example 12

Divide ( 2 − − 6) by ( + 2).

Solution

Step 1: Arrange the terms of the dividend and divisor in the order of
descending powers of a variable that appears in each term.

+ 2 2 − − 6

Step 2: Divide the first term of the divisor and the first term of the dividend
to get the first term of the quotient.

( 2 ÷ = )
+ 2 − − 6

Step 3: Multiply the divisor by the first term of the quotient.

( )( + 2) = 2 + 2
+ 2 − − 6

2 + 2

Step 3: Subtract the product from the dividend.

( 2 − − 6) − ( 2 + 2 ) = −3 − 6
+ 2 − −

− +

− 3 − 6

Mathematics Book 1 41

Step 4: Divide the first term of the product to the first term of the divisor.

− 3 (−3 ÷ = −3)
+ 2 2 − − 6

− 2 + 2

− − 6

Step 5: Multiply the divisor by the second term of the quotient.

− (−3)( + 2) = −3 − 6
+ 2 − − 6

− 2 + 2

− 3 − 6

− 3 − 6

Step 6: Subtract the first product to the second product.

−

+ 2 − − 6
− 2 + 2

− − −
− −

0 (−3 − 6) − (−3 − 6) = 0

Step 7: Repeat the steps 2, 3 and until there are no more terms left.

Therefore, the quotient from ( − − ) ÷ ( + ) is − .

To check the answer,
DIVISOR × QUOTIENT + REMAINDER = DIVIDEND

( + 2) × ( − 3) + 0 =? ( 2 − − 6)
( 2 − 3 + 2 − 6) + 0 =? ( 2 − − 6)
( 2 − − 6)
( 2 − − 6) =

Example 13

Find the quotient of (6 2 + 5 − 9) ÷ (3 + 4).

42 Mathematics Book 1

Solution

3 + 4 2 − 1
− 6 2 + 5 − 9
6 2 + 8
−
− 3 − 9

− 3 − 4

−5

Therefore, the quotient of ( + − ) ÷ ( + ) is − , = − .

Example 14

Divide ( 3 − 12 2 − 42) by ( 2 − 2 + 1). If the given
dividend has
Solution missing term/s,

− 10 add 0.
3 − 12 2 + 0 − 42
2 − 2 + 1 3 − 2 2 +
−
− 10 2 − − 42
− − 10 2 + 20 − 10

− 21 − 32

Therefore, the quotient is − , remainder − − .

To check,

( − 10)( 2 − 2 + 1) + (−21 − 32) = ( 2) + (−2 ) + (1) + (−10)( 2) +
=
= (−10)(−2 ) + (−10)(1) − 21 − 32
= 3 − 2 2 + − 10 2 + 20 − 42 − 21
3 − 12 2 − 42

Exercise 2G

1) Find the quotients of the following polynomials and check the answers.
a) ( 2 + 2 + 6) ÷ ( − 1)

________________________________ Check:
________________________________ _______________________
________________________________ _______________________
________________________________ _______________________
________________________________ _______________________
________________________________ _______________________

Mathematics Book 1 43

b) (2 2 + 7 + 6) ÷ ( + 2) Check:
________________________________ _______________________
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________________________________
Check:
c) (14 2 − 3 − 5) ÷ (7 − 5) _______________________
_______________________
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________________________________ Check:
________________________________ _______________________
_______________________
d) ( 3 + 6 2 + 7 + 2) ÷ ( + 1) _______________________
_______________________
________________________________ _______________________
________________________________ _______________________
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________________________________
________________________________ Check:
________________________________ _______________________
________________________________ _______________________
________________________________ _______________________
_______________________
e) (8 3 − 10 2 − + 1) ÷ ( − 1) _______________________
________________________________ _______________________
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________________________________

44 Mathematics Book 1

f) (2 3 − 5 2 − 8 + 15) ÷ ( − 3) Check:
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________________________________

g) (6 3 − 8 + 5) ÷ (2 − 4) Check:
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________________________________

h) (9 3 + 2 + 1) ÷ (3 + 1) Check:
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________________________________

Mathematics Book 1 45

i) ( 4 − 81) ÷ ( − 3) Check:
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j) (2 4 − 9 3 − 21 2 + 88 + 48) ÷ ( − 2)
Check:

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k) ( 4 − 3 2 + 4 + 5) ÷ ( 2 − + 1) Check:
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46 Mathematics Book 1

Summary

• Variable is any letters in the alphabet representing unknowns.
• Degree of monomial is the sum of the indices of variables.
• Coefficient is the number beside the variable.
• To add or subtract two or more monomials, they have to be similar.

➢ Two monomials are similar when they have the same variables with
the same indices.

• When multiplying monomial by another monomial, first MULTIPLY the
coefficients and then ADD the powers of the same variables.

• Multiplying monomial by a polynomial is to multiply each term of the
polynomial by the monomial using distributive property.

( + ) = +

• To divide monomial by a monomial, first DIVIDE the coefficients and then

SUBTRACT the powers of the same variables.

• Dividing a polynomial by a monomial is to divide each term of the polynomial

by a monomial.

+ + = + + ( ≠ )



• Polynomial is an algebraic expression consisting of constants and/or
monomial or two or more monomials. It is separated by one or more of the
four operations (+,×,÷, −), except NOT DIVISION BY A VARIABLE.

• The proper form of writing a polynomial is to put the monomial with the
highest degree first.

• There are two ways of multiplying polynomials by a polynomial:
➢ Multiplying polynomials in horizontal form by using the distributive
property.
➢ Multiplying polynomial in vertical form by lining up the polynomials as
you would for numerical multiplication.

• In dividing polynomials, use the long division in the way it is used for division
of integers.
➢ Arrange the terms of the dividend and the divisor in the order of
descending powers of a variable that appears in each. Divide the first
term of the divisor and the first term of the dividend to get the first
term of the quotient. Multiply the divisor by the first term of the
quotient. Subtract the product from the dividend. Repeat the steps
until there are no more terms left.

• To check the answer in division,

DIVISOR × QUOTIENT + REMAINDER = DIVIDEND

Mathematics Book 1 47

Revision Exercise

Section A (Skills : 1 – 3)

1) Given that = (2 2 + 5 − 1), = (−3 + 7), = (3 2 − 6 + 7)
and = (7 − 3), find the value of the following:

a) + = ______________________________________________
__________________________________________________________
__________________________________________________________

b) − = ______________________________________________
__________________________________________________________
__________________________________________________________

c) + − = _________________________________________
__________________________________________________________
__________________________________________________________

d) − ( − ) = _________________________________________
__________________________________________________________
__________________________________________________________

e) (− + ) + (− + ) = _______________________________
__________________________________________________________
__________________________________________________________

f) ( − ) − ( − ) = ____________________________________
__________________________________________________________
__________________________________________________________

2) Find the product.
a) 6 (6 ) = ______________________________________________

b) − (7 3 + 5 ) = ____________________________________

c) 4 (5 2 + 6 2) = _______________________________
__________________________________________________________

d) ( − )2 = _________________________________________
__________________________________________________________
__________________________________________________________

e) ( + )3 = _________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________

48 Mathematics Book 1

3) Find the quotient.

a) (8 − 4) ÷ 2 =

___________________________________________________________

___________________________________________________________

b) (3 5 − 2 4) ÷ 3 =

___________________________________________________________

___________________________________________________________

−15 4+5 7−25 5 =

c) 5 4

___________________________________________________________

___________________________________________________________

___________________________________________________________

d) ( 2 + 6 − 16) ÷ ( + 8) =
___________________________________________________________
___________________________________________________________
___________________________________________________________
___________________________________________________________
___________________________________________________________

Section B (Concepts : 4 – 14)
Circle the letter of the correct answer.

4) Which of the following is related to 6) If = + 1, = 2 − 2 and
the word monomial? = − 1, which equation is wrong?
a) Constant a) − + = − 2
b) Coefficient b) − = 0
c) Variable c) − =
d) Term d) 2 =

5) Which of the following statement is 7) What is the degree of the polynomial
not true about monomials? 2 2 3 − 2 + 6?
a) Constant is considered to be a
monomial. a) 6
b) It is the product of positive integer b) 3
powers of variables. c) 2
c) Monomials can be in fractional d) 5
form with variables on the
denominator. 8) Which expression is not a
d) Monomial is a polynomial with
one term. polynomial?

a) −12
b) 2 + −1
c) 2 − 2 + 7

d) −50

Mathematics Book 1 49

9) What is the sum of (4 2 + 5 − 8) 13) The statements are true about the
and (3 2 − 6 + 1)? operations involving polynomials,
a) 7 2 + − 7 except,
b) 7 2 − − 7 a) Adding polynomials is the same
c) 7 2 + + 7 as the steps used in combining
d) 7 2 − + 7 like terms.
b) Simply use distributive property
10) Which expression is equivalent to to multiply a monomial times a
3 2 + 4 − 1? polynomial.
a) ( 2 + 3 − 1) − (2 2 + ) c) Dividing polynomial by a
b) (2 2 + 4 + 1) − (− 2 + 2) monomial is to divide each term
c) (3 2 − 8) + (4 2 + 7) by the monomial and it is
d) (2 2 + 4 + 1) + (− 2 + 2) simplified as individual fractions.
d) Subtracting polynomials is to
11) Which product is equivalent to write the opposite signs of the
6 2 − 7 + 2? minuend and add like terms.
a) (6 + 1)( − 7)
b) (2 − 1)(3 + 2) 14) Divide (5 2 + 8 + 4) by ( + 2).
c) (−6 + 1)(− + 7) a) 5 + 2, remainder 8
d) (−2 + 1)(−3 + 2) b) 5 − 2, remainder 8
c) 5 + 3, remainder 2
12) If = −2, then find the value of the d) 5 − 3, remainder 2
expression 3( 3 − 4).
a) −36
b) 36
c) −30
d) 30

Section C (Word Problems : 15 – 17)

Circle the letter of the correct answer.

15) A rectangular painting is twice as Which expression represents the
long as it is wide. The painting has a
3-inch wide frame. Let represent area you are coloring?
the painting’s width. Which product a) 3 2 − 6
gives the area of the painting and b) 8 − 12
frame? c) 6 − 3 2
a) (2 )( )
b) (2 + 3)( + 6) d) 4 − 6
c) (2 + 6)( + 3)
d) (2 + 6)( + 6) 17) The top of a rectangular box has an
area of 3 − 4 2 − 3 + 18 cm2.
16) You are coloring a rectangular
tabletop. The length of the tabletop Its length is − 3 cm. What is the
is 6 inches less than 3 times the
width. Let represents the width. width of the box?
Which a) 2 + 6
b) 2 − + 6
50 Mathematics Book 1 c) 2 + − 6
d) 2 − − 6