Factors of Algebraic
Expressions
07
By studying this chapter you will be able to achieve the following
competencies.
« Expressing algebraic expressions with four terms as a product of two
factors by grouping the terms in pairs and taking out the common
factors.
« Resolving quadratic expressions into factors correctly.
« Resolving into factors a difference of two squares.
To recollect what you have already learnt in grade 8, find the factors of the
following algebraic expressions.
Exercise 7.1
(ii) 3x2 - 5xy
(iv) 5x2 - 15xy - 20xy2
(vi) 8c2 - 6cd + 2c
(viii) 6p - 24p2 + 30p3
(i) 2k - 12
(iii) 2ab - 8a + 4a2
(v) 30y2 - 6y - 6
(vii) 12a3 - 36a2b - 24ab2
7.1 Factors of expressions with four terms.
p
q
x
A
B
y
C
D
The area of the rectangle shown in the figure is
equal to the sum of the areas of the rectangles
A,B,C, and D
Taking the parts together and finding the area.
The sum of the areas of parts A , B , C , D
The length of the whole figure
The breadth of the whole figure
The area of the whole figure
Therefore px + qx + py + qy
=
=
=
=
=
px + qx + py + qy
p+q
x+y
(p + q) (x + y)
(p + q) (x + y)
\ (p + q) and (x + y) are the factors of the expression px + qx + py+ qy
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By expanding (p + q) (x + y) you will realise the truth of it
(p +q) (x+y) = p(x + y) + q(x + y)
= px + py + qx + qy
The fact that,
px + py + qx + qy = (p + q) (x + y) can be obtained by grouping the terms
into pairs and taking out the common factor.
px + py + qx + qy
= p(x + y) + q (x + y)
= (x + y) (p + q)
Example 1
Find the factors of 3a - 6c + 2ak - 4ck. By grouping into pairs and taking the common
factors out the expression can be written as,
3a - 6c + 2ak - 4ck
= 3(a - 2c) + 2k(a - 2c)
= (a - 2c) (3 + 2k)
[(a - 2c) is a common factor]
The accuracy can be checked by expanding (a - 2c) (3 + 2k)
(a - 2c) (3 + 2k)
= a(3 + 2k) - 2c (3 + 2k)
= 3a + 2ak - 6c - 4ck
Example 3
Example 2
Find the factors of: x2 + xy - x - y
2
x + xy - x - y = x(x + y) - 1(x + y)
=(x+y )(x-1)
Find the factors of : c2 - 3c + bc - 3b
2
c - 3c + bc - 3b = c (c - 3) + b (c -3)
=(c-3)(c+b)
Exercise 7.2
Find the factors of the following expressions by grouping into pairs and taking out the
common factor. Check the accuracy by multiplying the factors.
1.
3.
5.
7.
9.
2. p 2 - pq + 3pr - 3qr
4. pr + pt - qr - qt
6. x 2 + 2xy - 3x - 6y
8. x 2 -3xy - 6x2 + 18y
10. k - k l - l + l
ab + ac + 2b + 2c
ax - ay - bx + by
2pq + 6ps - 5q -15s
2ab - 2ac + b - c
4 - 4a + c - ac
7.2 Factors of quadratic expressions.
The length of a side of the square ABCD shown in the figure is x units, while the length
of PB is 3 units. Also the length of DR is 2 units. Let us find the area of the rectangle
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A
The length of AP
The breadth of AR
The area of APQR
=x+3
x
=x-2
= (x + 3) (x - 2)
B
P
T
R
Q
2
D
x
C
3
S
= The area of APSD - the area of SDRQ
= x(x + 3) - 2(x + 3)
= x2 + 3x - 2x - 6
= x2 + x - 6
This area could also be obtained in the following way.
The area of APQR
= The area of ABTR + the area of BPQT
= x(x - 2) + 3(x - 2)
2
= x - 2x + 3x - 6
2
= x +x-6
The area of APQR
When all these expressions are considered,
x2 + x - 6 = (x + 3) (x - 2)
2
Accordingly the factors of x + x - 6 are (x + 3) and (x - 2)
2
2
In the algebraic expression x + x - 6, the product of the term x and the constant
2
2
term gives -6x . The linear factors of -6x are as follows.
-6x2
→ 3x + (-2x) = x
For the algebraic sum of the pair of factors to be + x, which is the middle term of the
expression, the pair of factors that should be selected is + 3x and -2x.
2
2
Then the expression x + x - 6 can be written as x + 3x - 2x - 6
2
2
x +x-6
= x + 3x - 2x - 6
= x(x + 3) -2(x + 3)
= (x + 3) (x - 2)
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Example 4
Find the factors of y2 + 7y + 12
(+1y) ´ (+12y)
(-1y) ´ (-12y)
(+2y) ´ (+ 6y)
(-2y) ´ (- 6y)
+3y ´ + 4y
(-3y) ´ (-4y)
2
+ 12y
}
2
The product of the y term and the constant term
2
is + 12y
The pair of factors that has the middle term of the
expression + 7y as its sum is +3y and + 4y only.
Therefore
2
y +7y + 12
2
= y + 3y + 4y + 12
= y(y + 3) + 4(y + 3)
= (y + 3) (y + 4)
Example 5
Find the factors of a2 - 2a - 24
}
-24a2
2
The product of the term a
2
term is - 24a
and the constant
Out of these pairs of factors the suitable pair which
leads to the middle term - 2a is + 4a and -6a
a2 - 2a - 24
= a2 + 4a - 6a - 24 (the expression should be formed as shown here)
= a(a + 4) - 6(a + 4)
= (a + 4) (a - 6)
Example 6
2
Find the factors of : 30 - 17k + k
The product of the constant term and the k2 term is +30k2
2
Out of the pairs of factors of + 30k the suitable pair which leads to the middle term
-17k is the pair of factors -2k and -15k.
Accordingly
30 - 17k + k2
= 30 - 2k -15k + k2
(the expression should be formed as shown here)
= 2(15 - k) - k(15 - k) (2 and -k are common factors)
= (15 - k) (2 - k)
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When a quadratic expression is given with a common factor, the common factor should
be taken out first. Then the quadratic expression within brackets should be resolved
into factors.
Example 7
Find the factors of : 18+15a - 3a2
18 + 15a - 3a2
= 3(6 + 5a - a2)
The product of the a2 term and the constant term is -6a2.
2
Out of the factors of -6a , the correct pair which leads to the middle term + 5a of the
expression is -a and + 6a
Therefore
18 + 15a - 3a2
= 3[6 + 5a - a2]
= 3[6 + 6a - a - a2]
= 3[6(1 + a) - a(1 + a)]
= 3[(1 + a) (6 - a)]
= 3(1 + a) (6 - a)
Exercise 7.3
Find the factors of the following quadratic expressions. Check the accuracy of the
factors by writing the product of the factors.
1. a2 + 8a + 12
2. y2 + 3y - 18
3. p2 - 3p - 40
2
2
4. q - 11q + 24
5. r - r - 30
6. l 2 - 19l + 18
7. s 2 + 3s - 70
8. c2 + 9c + 20
9. 36 + 15k + k 2
10. 16 + 6x - x2
11. 30 - 7c - c2
12. 45 - 18y + y 2
2
2
13. 24 + 23x - x
14. 42 - 11z - z
15. 54 + 15d - d 2
16. 54 - 15f +f 2
17. 3x2 - 24x + 36
18. 45 + 30y + 5y 2
19. 72 - z - z2
20. 48 - 14g + g2
7.4 Factors of a difference of two squares.
A
P
B
b
Q
R
D
a
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C
The given diagram shows how a square APQR,
the length of a side of which is 'b', is placed inside
another square ABCD, the length of a side of
which is 'a'. Let us calculate the area of the shaded
portion.
When the difference of the areas of the squares is
considered the area of the shaded portion = a2 - b2
When the shaded portion is separated into two
parts
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by cutting along CQ and rearranging as shown here, a rectangle of B
length (a + b) and of breadth (a - b) is obtained.
P
bQ
C
Accordingly,
the area of the shaded portion = (a + b) (a - b)
a
Therefore
2
2
a - b = (a + b) (a - b)
C
Q
R
D
a-b
Accordingly,
2
2
the factors of a - b are (a + b) and (a - b)
(a + b) (a - b)
2
2
= a - ab + ab - b
2
2
=a -b
You will see that the product of the two binomial expressions above is a2 - b2
Example 8
Example 9
2
Find the factors of : x - 25
2
x - 25
2
2
=x - 5
= (x + 5) (x - 5)
2
2
Find the factors of : 16x - 9y
2
2
16x - 9y
2
2
= (4x) - (3y)
= (4x + 3y) (4x - 3y)
Example 10
Example 11
2
Find the factors of : 1 - 100p
2
1 - 100p
2
2
= 1 - (10p)
= (1 + 10p) (1 - 10p)
2
Find the factors : 3 - 12q
2
3 - 12q
= 3(1 - 4q2)
2
2
= 3[1 - (2q) ]
= 3(1 + 2q) (1 - 2q)
The problem shown in example 11 can also be resolved into factors as shown below.
2
3 - 12y
= 3(1 - 4y2)
The product of the y2 term and the constant term of the
expression within bracket is - 4y2. As there is no term
with 'y' in the expression, the factors should be selected
such that the algebraic sum of the factors of -4y2 is zero.
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The algebraic sum of the factors is zero only with the factors +2y and -2y.
Therefore
3[1 - 4y2]
= 3 [1 + 2y -2y - 4y2]
= 3 [1(1 + 2y) - 2y (1 + 2y)]
= 3 [(1 + 2y) (1 - 2y)]
= 3 (1 + 2y) (1 - 2y)
Exercise 7.4
Find the factors of the following expressions.
2
2
1. y - 9
2. p - 36
2
2
2
4. 4 - 9k
5. 4x -36y
2
2
7. 18c - 2
8. 4z - 100
2
3
10. 27d - 48
11. 3x - 243x
3.
6.
9.
12.
2
25 - a
2 2
a b -1
2
125k - 5
2
2
5m - 3125n
When finding factors; changing the position of the terms of an expression
should be done such that its accuracy is preserved.
Example 12
In the expression, ax + by - ay - bx, there are no factors common to the first two
terms and no factors common to the last two terms.
When rewritten by interchanging the second, third and fourth terms or writing the
st
1 term at the end it is
ax - ay - bx + by
by -ay -bx+ax
= a(x - y) -b(x - y)
= y(b - a) -x(b - a)
= (x - y) (a - b)
= (b - a) (y - x)
Example 13
In the expression pq - 6 + 3q - 2p there are no factors common to either the first two
terms or the last two terms. When rewritten with the third term in the second place and the
second term in the fourth place, the expression can be resolved into factors
pq + 3q - 2p - 6
= q(p + 3) - 2(p + 3)
= (p + 3) (q - 2)
Example 14
2
Find the factors of: x -12 + x
2
When rewritten with the third term in the first place it will be x + x -12
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-12x2
(x)
(-x )
(2x)
(-2x)
(+3x)
-3x
The pair of factors which leads to the middle
´ (-12x)
term +x are -3x and +4x
´ (+12x)
´ (- 6x)
x2 + x - 12
2
´ (+ 6x)
= x + 4x-3x-12
´ (- 4x)
= x(x + 4) -3(x + 4)
´ (+ 4x) → -3 x + 4x = x
= (x + 4) (x - 3)
Example 15
Find the factors of : - 4(3y - 5) + y
- 12y + 20 + y2
2
= y - 12y + 20
= y2 - 10y - 2y + 20
= y(y - 10) - 2(y - 10)
= (y - 10) (y - 2)
2
Exercise 7.5
Find the factors of the following algebraic expressions.
2
2
2
1. px - 1- x + p
2. 4 - k - 3k
2
3. ax - by + ay - bx
4. 3y - 28 + y
3
2
3
2
5. x + 2 + 2x + x
6. x + 1 + x + x
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