1. FACTORING ALGEBRAIC EXPRESSIONS
Created by ﺟﻴﻄ for mathlabsky.wordpress.com
Created by ﺟﻴﻄ for mathlabsky.wordpress.com

2. P Q P + Q P – Q P x Q FACTORING ALGEBRAIC EXPRESSIONS
I. Arithmetical operations on algebraic :
Adding
Substracting
Multiplying
Dividing
Exponentiating P Q
P + Q
P – Q
P x Q
2x – 3
4x + 1
6x -2
-2x – 4
8x x – 3
x + 2
2x – 1
3x + 1
-x + 3
2x2 + 3x – 2
2a – a2 2a
4a – a2
-a2
4a2 – 2a3
2(3 – a)
1 – a
7 – 3a
5 – a
6 – 8a + 2a2
4b - 1
(2b + 1) 6b
2b – 2
8b2 + 2b – 1

3. P Q P : Q P P2 P3 4a 2 2a 4a2 6a3 3a 2a2 12a – 4a2 6 – 2a 8a + 16 4
-3ab3
9a2b6
-27a3b9
4ab
16a2b2
64a3b3

4. II. Factoring Algebraic Expressions
12 ab2 – 9b3c2
= 3b2 (4a – 3bc2)
4x = 22.x
12ab2 = 22.3.a.b2
8 = 23
9b3c2 = 32.b3.c2
GCD = 22 = 4
GCD = 3.b2
Exercise
12xy2 + 4x2y3z
8x – 12y
24xyz2 + 9x2y
10xy3 + 2y2z
1. 4xy2 (3 + xyz)
2. 4 (2x - 3y)
3. 3xy (8z2 + 3x)
4. 2y2 (5xy + z)

5. III. Special form
A. x2 + 2xy + y2
= (x + y) 2
B. x2 - 2xy + y2
= (x - y) 2
(x + y) 2
=(x + y)(x + y)
(x - y) 2
=(x - y)(x - y)
= x2 + xy + xy + y2
= x2 - xy - xy + y2
= x2 + 2xy + y2
= x2 - 2xy + y2
x2 + 4x + 4 = ( … ) 2
( x + 2) 2
x2 - 6x + 9 = ( … ) 2
( x - 3) 2

6. A. x2 + 2xy + y2 = (x + y) 2 B. x2 - 2xy + y2 = (x - y) 2 a2 + 4a + 4
Factor the following algebraic expressions!
a2 + 4a + 4
16x2 – 24x + 9
4a2 – 4ab + b2
9m2 + 30mn + 25n2
25p2 + 70pq + 49q2
1. (a + 2)2
2. (4x - 3)2
3. (2a - b)2
4. (3m + 5n)2
5. (5p + 7q)2

7. a2 – b2 (2m) 2 – 32 x2 – 49 m2 – 121 64 – y2 1. (a – b)(a + b)
C. x2 - y2 C.
= (x - y)(x + y)
(x – y )(x + y)
= x2 + xy – xy - y2
= x2 - y2
a2 – b2
(2m) 2 – 32
x2 – 49
m2 – 121
64 – y2
1. (a – b)(a + b)
2. (2m – 3)(2m + 3)
3. (x – 7 )(x + 7)
4. (m – 11 )(m + 11)
5. (8 – y)(8 + y)

8. D. Factoring ax2 + bx + c, when a = 1
x2 + bx + c = (x + p)(x + q)
Where c = p x q and b = p + q
Example :
1. x2 + 10x + 16
====> a = 1, b = 10, c = 16
p = …? q = …?
… x … = 16
… + … = 10 8 2 8 2
x2 + 10x + 16
 (x + 8)(x + 2)
2. x2 – x – 6
====> a = 1, b = -1, c = -6
p = …? q = …? -3
… x … = -6
… + … = -1 2
x2 – x – 6  (x – 3)(x + 2) -3 2

9. a2 + 5a + 6 a2 + a – 6 (a + 3)(a + 2) (a – 2 )(a + 3) y2 + 6y + 9
Factor the following algebraic expressions!
a2 + 5a + 6
a2 + a – 6
y2 + 6y + 9
y2 – 14y + 24
p2 + 4p – 5
(a + 3)(a + 2)
(a – 2 )(a + 3)
(y + 3)(y + 3)
(y – 12)(y – 2)
(p + 5)(p – 1)

10. E. Factoring ax2 + bx + c, when a ≠ 1
p + q = b
p x q = a x c
Example :
1. 2x2 + 7x + 6
===> a = 2, b = 7, c = 6
p = …? q = …?
… + … = 7
… x … = 12 4 3 4 3
2x2 + 7x + 6
 2x2 + 4x + 3x + 6
 2x2 + 4x + 3x + 6
 2x(x + 2) + 3(x + 2)
 (x + 2) (… + …)
(2x + 3)

11. 2. 6x2 + 13x - 5
===> a = 6, b = 13, c = -5
p = …? q = …?
… + … = 13
… x … = -30 -2 15 -2 15
6x2 + 13x - 5
 6x2 - 2x + 15x - 5
 6x2 - 2x + 15x - 5
 2x(3x - 1) + 5(3x - 1)
 (3x - 1) (… + …)
(2x + 5)

12. IV. OPERATIONS ON ALGEBRAIC FRACTIONS
Example: