This document provides examples of factorizing polynomials using various factorization techniques. It contains 30 questions with step-by-step solutions demonstrating how to factorize expressions involving single variables, binomials, trinomials, and polynomials using formulas like difference of squares, perfect square trinomials, and grouping. The techniques shown include finding common factors, using factor trees, recognizing patterns, and applying factorization identities.

1. Curriculum on factorization
Factorization a2-b2, trinomial with
perfect square, trinomial in the form
of ax2  bx c.
PRESENTED BY
MANINATH NEUPANE
CLASS 8

2. Factorization

3. Q 1.Factorise: 2ax+4ay
ax ay
a
x
a
y
a
x
a
x
a
y
a
y
a
y
a
y
2a
(x+2y)

4. Q 1.Factorise: 2ax+4ay
=2 .
=2.2
2ax + 4aySolution:
2ax
4ay
=
a . x
. a . y
( + )
Rule 1. We have to take common
factors as common and remaining
factors should be kept inside
parenthesis.

5. Q 2.Factorise: 4a2b-6ab2+10ab
(
4a2b-6ab2+10ab
4a2b
6ab2
10ab
=2. 2. a. a. b
2. 3. a. b. b=
2. 5. a. b=
- + )
Solution:
=
Rule 1. We have to take common factors as common
and remaining factors should be kept inside
parenthesis.

6. Q 3. Resolve into factors: 2x (a + b)- 3y (a + b)
Solution:
2x (a + b)- 3y (a + b) 2x (a + b)=
3y (a + b)=
2 . x . (a+b)
3 . y . (a+b)
= ( - )
Rule 1. We have to take common factors as common
and remaining factors should be kept inside
parenthesis.

7. Q 4.Factorise: a2-ax+ab-bx
Solution: a2-ax + ab-bx
= (
a2= a . a
ax= a . x- ) + b(a - x)
=(a-x)(a+b)
Rule 2. We have to arrange the terms in groups such
that each group has common factor.

8. Q 5.Factorise: a(x2-y2)+x(y2-a2)
Solution: a(x2-y2)+x(y2-a2)
=ax2-ay2+xy2-xa2
=ax2+xy2-ay2-xa2
=x(ax+y2)-a(y2+xa)
=x(ax+y2)-a(xa+y2)
=(ax+y2)(x-a)
We have to
simplify
the
expression
first so that
we can
factorize.

9. Q 6.Factorise: x2-9
Solution:
x 2 - 9
= x 2 – 3 2
∴ x2 – 32 = a 2 – b 2
Let x=a and 3=b
= (a – b)(a + b)
= ( x – 3 )( x + 3 )
a 2 – b 2 = ( a – b ) ( a + b )
Putting the value of a and b.
Rule 3. We have to
use the suitable
formula.

10. Q 6.Factorise: x2-9
Solution:
x 2 - 9
= x 2 – 3 2
= (
x
–
3
)(
x
+
3
)
a 2 – b 2 = ( a – b ) ( a + b )
Rule 3. We have to
use the suitable
formula.

11. Q 6.Factorise: x2-9
Solution:
x 2 - 9
= x 2 – 3 2
= (
x
–
3
)(
x
+
3
)
a 2 – b 2 = ( a – b ) ( a + b )
Rule 3. We have to
use the suitable
formula.

12. Q 7.Factorise: 25x2-9y2
Solution: 25x2-9y2
a2 –b2 =(a–b)(a+b)= (5x)2 –(3y)2
= (
5x
–
3y
)(
5x
+
3y
)
Rule 3. We have to
use the suitable
formula.

13. Q 8.Factorise: a4-16
Solution: a4-16
=(a2)2-42
=(a2-4)(a2+4)
=(a2-22)(a2+4)
=(a-2)(a+2)(a2+4)
a2 –b2 =(a–b)(a+b)
Rule 3. We have to
use the suitable
formula.

14. Q 9.Factorise: (x2+y2)2-x2y2
Solution: (x2+y2)2-x2y2
=(x2+y2)2-(xy)2
={(x2+y2)-(xy)}{(x2+y2)+(xy)}
=(x2+y2-xy)(x2+y2+xy)
=(x2-xy+y2)(x2+xy+y2)
a2 –b2 =(a–b)(a+b)
Rule 3. We have to
use the suitable
formula.

15. Q 10.Factorise: 4-(a-b)2
Solution: 4-(a-b)2
=22-(a-b)2
={2-(a-b)}{2+(a-b)}
=(2-a+b)(2+a-b)
a2 –b2 =(a–b)(a+b)
Rule 3. We have to
use the suitable
formula.

16. Q 11. Simplify by factorization process: 722-622.
Solution: 722-622
=(72-62)(72+62)
=10×134
=1340
We have to use the
suitable formula.

17. Q 12. Simplify by factorization process: 101×99
Solution: 101×99
= (100+1)(100-1)
= (10000-1)
= 9999
We have to use the
suitable formula.

18. Q 13. If a+b=8 and ab =15, find the value of a and b.
numbers(a & b) sum(a+b) = 8 product(a.b) =15
3+5=8
1.7=7
∴Numbers are 3 &5.
When a=3 then b=5
and when a=5 then b=3.
1 and 7
2 and 6
3 and 5
1+7=8
2+6=8 2.6=12
3.5=15
Solution:
We use hit and trial
method.

19. Q 14. If a+b=-10 and ab=24,find the value of a and b.
numbers(a & b) a.b=24 a+b=-10
-3.-8=24
-1-24=-25
∴Numbers are -4 & -6.
When a=-4 then b=-6
and when a=-6 then b=-4.
-1 and -24
-2 and -12
-3 and -8
-1.-24=24
-2.-12=24 -2-12=-14
-3-8=-11
Solution:
-4 and -6 -4.-6=24 -4-6=-10
We use hit and trial
method.

20. Q 15. Factorize: x2+5x+6
x2 1x 1x 1x 1x 1x
1
1
1
1
1
1
x+3
x+2
∴ x2+5x+6=(x+3)(x+2)
What did we do here?

21. Q 15. Factorize: x2+5x+6
Solution: x2+5x+6
Rule 4. Method
1. Multiply coefficient of x2
and constant term.
1×6=6(product)
2. Find the possible factors of
the product 6.
6=1×6
6=2×3
3. Remember the sign of
constant term.
= x2+(3+2)x+6
= x2+3x+2x+6
= x(x+3)+2(x+3)
= (x+3)(x+2)
4. If sign of constant term is +
then choose the pair of factors
whose sum is 5 (coefficient of x)

22. Q 16. Resolve into factors:
x2-9x+20
Rule 4. Method
1. Multiply coefficient of x2 and
constant term.
1×20=20(product)
2. Find the possible factors of the
product 20.
20=1×20
20=2×10
4. If sign of constant term is + then
choose the pair of factors whose sum is
9 (coefficient of x)
20=4×5
Solution: x2-9x+20
= x2-(5+4)x+20
= x2-5x-4x+20
= x(x-5)-4(x-5)
= (x-5)(x-4)
3. Remember the sign of constant term.

23. Q 17. Factorize: x2+3x-18 Rule 4. Method
1. Multiply coefficient of x2 and
constant term.
1×18=18(product)
2. Find the possible factors of the
product 18.
18=1×18
18=2×9
18=3×6
Solution: x2+3x-18
=x2+(6-3)x-18
=x2+6x-3x-18
=x(x+6)-3(x+6)
=(x+6)(x-3) 4. If sign of constant term is - then
choose the pair of factors whose
difference is 3 (coefficient of x)
3. Remember the sign of constant term.

24. Q 18. Factorize: x2-5x-14 Rule 4. Method
1. Multiply coefficient of x2 and
constant term.
1×14=14(product)
2. Find the possible factors of the
product 14.
14=1×14
14=2×7
4. If sign of constant term is - then
choose the pair of factors whose
difference is 3 (coefficient of x)
3. Remember the sign of constant term.
Solution: x2-5x-14
=x2-(7-2)x-14
=x2-7x+2x-14
=x(x-7)+2(x-7)
=(x-7)(x+2)

25. Q 19. Factorize: 2x2+7x+3 Rule 4. Method
1. Multiply coefficient of x2 and
constant term.
2×3=6(product)
2. Find the possible factors of the
product 6.
6=1×6
6=2×3
4. If sign of constant term is + then
choose the pair of factors whose sum is
7 (coefficient of x)
3. Remember the sign of constant term.
Solution: 2x2+7x+3
=2x2+7x+3
=2x2+(6+1)x+3
=2x2+6x+1x+3
=2x(x+3)+1(x+3)
=(x+3)(2x+1)

26. Q 20. Factorize: x2+x-30
Solution: x2+x-30
Rule 4. Method
1. Multiply coefficient of x2 and
constant term.
1×30=30(product)
2. Find the possible factors of the
product 30.
30=1×30
30=2×15
4. If sign of constant term is - then
choose the pair of factors whose
difference is 1 (coefficient of x).
3. Remember the sign of constant term.
30=3×10
30=5×6
=x2+x-30
=x2+(6-5)x-30
=x2+6x-5x-30
=x(x+6)-5(x+6)
=(x+6)(x-5)

27. Q 21. Factorize: x2+4x+4
Solution:
x2+4x+4
=x2+2.x.2+22
=(x+2)2
Rule 3. We have to
use the suitable
formula.
Rule 4.
Or,

28. Q 22. Factorize: 4x2-12x+9
Solution:
4x2-12x+9
= (2x)2-2.2x.3+32
= (2x-3)2
Rule 3. We have to
use the suitable
formula.
Rule 4.
Or,

29. Q 23. Factorize: 4x – 8
Q 24. Factorize: 4x3 - 6x2 + 8x
Q 25. Factorize: xm + ym + xa + ya
Q 26. Factorize: a( a + 4 ) + 2 ( a + 4 )
Q 27. Factorize: a( a – 3 ) - 4( 3 – a )
Q 28. Factorize: – a – b + 1 + ab
Q 29. Factorize: x2y + xy2z + zx + yz2

30. Q 30. Factorize: x2 – 1
Q 31. Factorize: 4x2 – 9b2
Q 32. Factorize: 1-36x2
Q 33. Factorize: x2-36
Q 34. Factorize: 16a2 – 25b2
Q 35. Factorize: 72x3 – 50x
Q 36. Find the area of shaded part.
y
y
2
2

31. Q 37. Factorize: x2 + 10x +25
Q 38. Factorize: p2 – 24p+144
Q 39. Factorize: 9a2 – 66a + 121
Q 40. Factorize: x2 + 9x +18
Q 41. Factorize: x2 - 9x +18
Q 42. Factorize: x2 + 3x – 18
Q 43. Factorize: x2 - 3x – 18
Q 44. Factorize: 15x2 – x – 2
Q 45. Factorize: 125x3 + 8a3
Q 46. Factorize: (x – y )3 + 8(x + y)3