Useful
formulae
(i) (a + b)2 = a2 + 2ab +
b2
(ii) (a – b)2 = a2 – 2ab +
b2
(iii) a2 – b2 = (a + b) (a –
b)
(v) a3 + b3 = (a + b) (a2
– ab + b2)
(vi) a3
+ b3 = (a + b)3 – 3a(a + b)
(vii) a3 – b3 = (a – b) (a2
+ ab + b2)
(viii) a3
– b3 = (a – b)3 + 3ab (a – b)
(ix) (a + b)3 = a3 + 3a2b
+ 3ab2 + b3
(ix) (a
+ b)3 = a3 + b3 + 3ab (a + b)
(x) (a – b)3
= a3 – 3a2b + 3ab2 – b3
(xi) (a – b)3 = a3 – b3 – 3ab(a – b)
(xii) (a + b + c)2 = a2 + b2
+ c2 + 2ab + 2bc + 2ca
(xiii) (a – b – c)2 = a2 + b2
+ c2 – 2ab + 2bc + 2ca
(xiv) a4 + a2b2 + b4
= (a2 + ab + b2) (a2 – ab + b2)
Factorize
Factorization
The process
of finding factors is called factorization.
The factors
of 12 are 1, 2, 3, 4, 6 and 12
The prime
factors of 12 are 2, 2, 3
Factorization of an algebraic expression
The process
of finding factors of algebraic expression is called factorization of algebraic
expression.
3x2
= 3 ´ x ´ x
Things to know while doing factorization of an algebraic expression
ü Check
whether there is any common factor on each term. If same factor is there in
each term, then take that as common factor.
ü Rearrange the term, (if necessary) and take common.
ü Use formula if necessary
ü Split the term or Break the term
Q.1.
Factorize: x2 – 2x
Soln: Here, x2 – 2x = x(x – 2) Ans.
Q.2.
Factorize: 3a2 – 12ab
Soln: Here, 3a2 – 12ab = 3a(a – 4b) Ans.
Q.3. Factorize: ab + bc + ax + cx
Soln: Here, ab + bc + ax + cx = b(a + c) + x(a + c)
= (a + c)(b + x) Ans.
Q.4.
Write the factors of x2 – 25
Soln: Here, x2 – 25 = x2 – 52
= (x – 5)(x + 5) Ans.
Q.5. Factorize: x(x – y)2 + 2xy(x – y)
Soln: Here, x(x – y)2 + 2xy(x – y) = x(x – y) {(x – y) + 2y}
= x(x
– y) (x – y + 2y)
= x(x – y)(x + y) Ans.
Q.6. Factorize): 4a4 + x4
Soln: Here, 4a4
+ x4 = (2a2)2
+ (x2)2
=
(2a2 + x2)2 – 2.2a2.x2
=
(2a2 + x2)2 – 4a2x2
=
(2a2 + x2)2 – (2ax)2
=
{(2a2 + x2) – 2ax}{(2a2 + x2) +
2ax}
= (2a2 – 2ax + x2) (2a2 + 2ax + x2) Ans.
Q.7. Factorize: p4 + 64q4
Soln: Here, p4
+ 64q4 = (p2)2 + (8q2)2
= (p2 + 8q2)2
– 2p2.8p2
= (p2 + 8q2)2
– 16p2q2
= (p2 + 8q2)2
– (4pq)2
= (p2 + 8p2
– 4pq) (p2 + 8q2 + 4pq)
= (p2 – 4pq + 8q2) (p2 + 4pq + 8q2) Ans.
Q.9. Factorize: x4 + 5x2 + 9
Soln: Here, x4 + 5x2 + 9 = x4 + 9 + 5x2
= (x2)2 + 32 + 5x2
= (x2 + 3)2 – 2x2.3 + 5x2
= (x2 + 3)2 – x2
= {(x2 + 3) – x} {(x2 + 3) + x}
= (x2 – x + 3) (x2 + x + 3) Ans.
Q.10. Factorize: p2 + 2p – 8 – q2 – 6q
Soln: Here, p2 + 2p – 8 – q2 – 6q
= p2 + 2p + 1 – 9 – q2 –
6q
= p2 + 2.p.1 + 12 – (q2
+ 6q + 9)
= (p + 1)2 – (q2 + 2.q.3
+ 32)
= (p + 1)2 – (q + 3)2
= {(p + 1) – (q + 3)}{(p + 1) + (q + 3)}
= (p – q – 2) (p + q + 4) Ans.
Q.11. Factorize: 4m2 – 12m – 16 – 9x2 + 30n
Soln: Here, 4m2 – 12m – 16 – 9x2 + 30n
= 4m2 –12m + 9 – 25 – 9n2
+ 30n
= (2m)2 – 2.2m.3 + 32 –
(9x2 – 30n + 25)
= (2m – 3)2 – {(3n)2 –
2.3n.5 + 52)
= (2m – 3)2 – (3n – 5)2
= {(2m – 3) – (3n – 5)}{(2m – 3) + (3n – 5)}
= (2m – 3n + 2) (2m + 3n – 8) Ans.
Q.12.
Factorize: a2 – 6a + 8 – b2 + 2b
Soln: Here, a2 – 6a + 8 – b2 + 2b
= a2 – 6a + 9 –
1 – b2 + 2b
= a2 – 2.a.3 + (3)2 – (b2
– 2b + 1)
= (a – 3)2 – (b – 1)2
= {(a – 3) + (b – 1)}{(a – 3) – (b – 1)}
= (a + b – 4) (a – b – 2) Ans.
Q. 13. y4 + y2 + 1 [Use the formula a2 + b2 = (a + b)2 - 2ab or, (a - b)2 + 2ab]
Solution
Here,
y4 + y2 + 1 = y4 + 1 + y2
= (y2)2 + (1)2 + y2
= (y2 + 1)2 - 2.y2.1 + y2
= (y2 + 1)2 - 2y2 + y2
= (y2 + 1)2 - y2
= {(y2 + 1) + y}{(y2 + 1) - y}
= (y2 + y + 1)(y2 - y + 1) Ans
Q.13. x4 + x2y2 + y4
Solution:
Here, x4
+ x2y2 + y4 = x4 + y4 + x2y2
= (x2)2 +
(y2)2 + x2y2
= (x2 + y2)2
– 2.x2.y2 + x2y2
= (x2 + y2)2
– 2x2y2 + x2y2
= (x2 + y2)2
– x2y2
= (x2 + y2)2
– (xy)2
= {(x2 + y2)
+ xy} {(x2 + y2) - xy}
= (x2 + xy + y2)(x2 - xy + y2) Ans.
Q.14. x8 + x4 + 1
Solution
Here,
x8
+ x4 + 1 = x8 +1 + x4
=
(x4)2 + (1)2 + x4
=
(x4 + 1)2 - 2.x4.1 + x4
=
(x4 + 1)2 - 2x4 + x4
=
(x4 + 1)2 - x4
=
(x4 + 1)2 - (x2)2
=
{(x4 + 1) + x2}{(x4 + 1) - x2}
=
(x4 + 1 + x2)(x4 - x2 + 1)
=
{(x2)2 + (1)2 + x2}(x4 - x2 + 1)
=
{(x2 + 1)2 - 2x2.1 + x2}(x4 - x2 + 1)
=
{(x2 +1)2 - x2}(x4 - x2 + 1)
=
{(x2 +1) + x}{(x2 +1) - x} (x4 - x2
+ 1)
=
(x2 + x + 1)(x2 - x + 1} (x4 - x2 + 1)
Q. 15. x4 + 4y4
Solution a2 + b2 = (a +
b)2 - 2ab or (a - b)2 + 2ab
Here, x4 + 4y4 = (x2)2 + (2y2)2
= (x2 + 2y2)2
– 2.x2.2y2
= (x2 + 2y2)2
– 4x2y2
= (x2 + 2y2)2
– (2xy)2
= {(x2 + 2y2)
+ 2xy} {(x2 + 2y2) - 2xy)
= (x2 + 2xy
+ 2y2) (x2 - 2xy + 2y2)
Q. 16. 81x4 + 64y4
Solution
Here, 81x4 + 64y4 = (9x2)2
+ (8y2)2
= (3x2
+ 8y2)2 – 2.9x2.8y2
= (3x2
+ 8y2)2 – 144x2y2
= (3x2
+ 8y2)2 – (12xy)2
= {(3x2
+ 8y2) + 12xy} {(3x2
+ 8y2) - 12xy)
= (3x2
+ 12xy + 8y2) (3x2 - 12xy + 8y2)
Q.17. x4 - 7x2 + 1
Solution
Here, x4 - 7x2 + 1 = x4 + 1 - 7x2
=
(x2)2 + (1)2 - 7x2
=
(x2 + 1)2 - 2.x2.1 - 7x2
=
(x2 + 1)2 - 9x2
=
(x2 + 1)2 - (3x) 2
=
{(x2 + 1) + 3x}{(x2 + 1)- 3x}
=
(x2 + 3x + 1)(x2 - 3x + 1)
Q. 18. x4 - 5x2y2 + 4y4
Solution
Here, x4 - 5x2y2 + 4y4 = x4 + 4y4 - 5x2y2
=
(x2)2 + (2y2)2 - 5x2y2
=
(x2 + 2y2)2 - 2.x2.2y2 - 5x2y2
=
(x2 + 2y2)2 - 4x2y2 - 5x2y2
=
(x2 + 2y2)2 - 9x2y2
=
(x2 + 2y2)2 - (3xy)2
=
{(x2 + 2y2) + 3xy} {(x2 + 2y2) - 3xy}
=
(x2 + 3xy + 2y2)(x2 - 3xy + 2y2)
Q. 19. 49x4 - 154x2y2 + 9y4
Solution
Here, 49x4 - 154x2y2 + 9y4 = 49x4
+ 9y4 - 154x2y2
=
(7x2)2 + (3y2)2 - 154x2y2
=
(7x2 + 3y2)2 - 2.7x2.3y2- 154x2y2
=
(7x2 + 3y2)2 - 196x2y2
23. x4 - 8x2 - 33 - 14y - y2
Solution: x4 - 8x2 - 33 - 14y - y2 = (x2)2 - 2.x2.4 + (4)2 - (4)2 - 33 - 14y - y2
= (x2 - 4)2 - 16 - 33 - 14y + y2
= (x2 - 4)2 - 49 - 14y - y2
= (x2 - 4)2 - (49 + 14y + y2)
= (x2 - 4)2 - {(7)2 + 2.7.y + y2)}
= (x2 - 4)2 - (7 + y)2
= {(x2 - 4) + (7 + y)} {(x2 - 4) - (7 + y)}
= (x2 - 4 + 7 + y) (x2 - 4 - 7 - y)
= (x2 + 3 + y) (x2 - 11 - y)
24. x4 - 6x2 - 7 - 8x - x2
Solution
Here, x4 - 6x2 - 7 - 8x - x2
= (x2)2 - 2.x2.3 + (3)2 - (3)2 - 7 - 8x - x2
= (x2 - 3)2 - 9 - 7 - 8x - x2
= (x2 - 3)2 - 16 - 8x - x2
= (x2 - 3)2 - (16 + 8x + x2)
= (x2 - 3)2 - {(4)2 + 2.4.x + x2}
= (x2 - 3)2 - (4 + x)2
= {(x2 - 3) + (4 + x)} {(x2 - 3) - (4 + x)}
= (x2 - 3 + 4 + x) (x2 - 3 - 4 - x)
= (x2 + x + 1) (x2 - x - 7) Ans
25. x4 - 12x2 - 28 + 16y - y2
Solution
Here, x4 - 12x2 - 28 + 16y - y2
= (x2)2 - 2.x2.6 + (6)2 - (6)2 - 28 + 16y - y2
= (x2 - 6)2 -36 - 28 + 16y - y2
= (x2 - 6)2 - 64 + 16y - y2
= (x2 - 6)2 - (64 - 16y + y2)
= (x2 - 6)2 - {(8)2 - 2.8.y + y2}
26. x4 - 10x2 + 24 + 6y2 - 9y4
Solution
Here, x4 - 10x2 + 24 + 6y2 - 9y4
= (x2)2- 2.x2.5 +
(5)2 - (5)2 + 24 + 6y2- 9y4
= (x2 - 5)2 - 25 + 24 + 6y2- 9y4
= (x2 - 5)2 - 1 + 6y2- 9y4
= (x2 - 5)2 - (1 - 6y2- 9y4)
= (x2 - 5)2 - {(1)2 - 2.1.3y2+
(3y2)2}
= (x2 - 5)2 - (1 - 3y2)2
27. x2- 10xy + 16y2 - z2 + 6yz
Solution
Here, x2
- 10xy + 16y2
- z2 + 6yz
= x2 - 2.x.5y + (5y)2
- (5y)2 + 16y2
- z2 + 6yz
= (x2- 5y)2 - 25y2 + 16y2
- z2 + 6yz
= (x2 - 5y)2 - 9y2 - z2 + 6yz
= (x2- 5y)2 - (9y2+ z2
- 6yz)
= (x2- 5y)2 - (9y2 - 6yz+ z2)
= (x2- 5y)2 - {(3y)2 - 2. 3y . + z2)
28. 4225x4- 130x2 - 3 + 36y2 - 81y4
Solution
Here,
4225x4- 130x2 - 3 + 36y2 - 81y4
= (65x2)2 - 2.65x2. 1 +
(1)2 - (1)2 - 3 + 36y2 - 81y4
= (65x2 - 1)2 - 1 - 3 + 36y2 - 81y4
= (65x2 - 1)2 - 4 + 36y2 - 81y4
= (65x2 - 1)2 - (4 - 36y2 + 81y4)
= (65x2 - 1)2 - {(2)2 - 2.2.9y2 + (9y2)2}
= (65x2 - 1)2 - (2 - 9y2)2
= {(65x2 - 1) + (2 - 9y2)} {(65x2
- 1) - (2 - 9y2)}
= (65x2 - 1 + 2 - 9y2) (65x2
- 1 - 2 + 9y2)
= (65x2 + 1 - 9y2) (65x2
- 3 + 9y2)
Ans
29. x2 - 90xy + 2050y2 - 550yz + 3025z2
Solution
Here, x2 - 90xy + 2050y2 - 550yz + 3025z2
= x2 -
2.x.45y + (45y)2 - (45y)2 - 2050y2 -
550yz -
3025z2
= (x - 45y)2 -
2025y2 + 2050y2 - 550yz -
3025z2
= (x - 45y)2 + 25y2
-
550yz + 3025z2
= (x - 45y)2 -
{(5y)2 -
2 .5y . 55yz + (55z)2}
= (x - 45y)2 - (5y - 55z)2
= {(x - 45y)
+ (5y -
55z)} {(x - 45y) - (5y -
55z)}
= (x - 45y + 5y - 55z)(x - 45y - 5y +
55z)
= (x - 40y - 55z)(x - 50y + 55z)
30. 169x2 - 52x - 56y - 196y2
Solution
Here,
169x2 - 52x - 56y - 196y2
= (13x)2 - 2.13x .2 + (2)2
- (2)2 - 52x - 56y - 196y2
= (13x - 2)2- 4 - 56y - 196y2
= (13x - 2)2- (4 +
56y + 196y2)
= (13x - 2)2- {(2)2 + 2.2 14y + (14y)2}
= (13x - 2)2- (2 + 14y)2
= Complete it
31. 4225x2 - 130xy - 3y2 - z2 - 4yz
Solution
Here,
4225x2 - 130xy - 3y2 - z2 - 4yz
= (65x)2 - 2. 65x. y + y2
- y2 - 3y2 - z2 - 4yz
= (65x - y)2 - 4y2 - z2 - 4yz
= (65x - y)2 - (4y2 + z2 + 4yz)
= (65x - y)2 - {(2y)2 + 2.2y.z + z2)}
= (65x - y)2 - (2y + z)2
= Complete
it
32. 289x2
+ 170x + 24 -
38y2 -361y4
Solution
Here, 289x2
+ 170x + 24 - 38y2 -361y4
= (17x)2 + 2.17x.5 + (5)2
- (5)2 +
24 - 38y2 -361y4
= (17x + 5)2 - 25 + 24 - 38y2 -361y4
= (17x + 5)2 - 1 - 38y2 -361y4
= (17x + 5)2 - (1 + 38y2 +361y4)
= (17x + 5)2 - {(1)2 + 2.1.19y2 +(19y2)2}
= (17x + 5)2 - (1 + 19y2)2
=
Complete it
33. x2
+ 50xy + 641y2 -
8yz -z2
Solution
Here, x2
+ 50xy + 641y2 - 8yz -z2
=
x2 + 2.x.25y + (25y)2
- (25y)2 +
641y2 - 8yz -z2
=
(x + 25y)2 - 625y2 + 641y2 - 8yz -z2
=
(x + 25y)2 - 16y2 - 8yz -z2
=
(x + 25y)2 - (16y2 + 8yz + z2)
=
(x + 25y)2 - {(4y)2 + 2.4y.z + z2)}
=
(x + 25y)2 - (4y + z)2
= Complete
it
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