Factorization |
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- From the given expression, first observe its form. Then use the appropriate formula and obtain the factors.
- We derive formulae for some products. We are given the product of two factors and we need to find those factors.
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Expression |
Factors |
a2 + 2ab + b2 |
( a + b) 2 |
a2- 2ab + b2 |
( a - b) 2 |
a2 - b2 |
( a + b ) ( a - b ) |
a3+ b3 |
( a + b ) ( a2 - a b + b2 ) |
a2 - b2 |
( a - b ) ( a2+ a b + b2 ) |
a3+ 3a2b + 3ab2 + b3 |
( a + b )3 |
a3 - 3a2 b + 3ab2- b3 |
( a - b )3 |
a2+ b2+ c2 + 2ab + 2bc + 2ca |
( a+ b + c) 2 |
a3+ b3 + c3 - 3abc |
( a + b + c ) ( a2+ b2 + c2 - a b - b c - ca ) |
x2+ x (a + b ) + ab |
( x + a ) ( x + b ) |
x3+ x2 ( a + b + c ) |
( x + a ) ( x + b ) ( x + c ) + x ( a b + b c + c a ) + abc |
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Factorize each of the following polynomials |
Example 1 |
x2 + 6x + 9
Solution:
Using the product with a = x, b = 3, we have
x2 + 6x + 9 = ( x + 3) 2 |
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Example 2 |
1 - 8x + 16x2
Solution:
Using the product with a = 1, b = 4x, we have
1 - 8x + 16x2= ( 1 - 4x) 2 |
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Example 3 |
4x2 - 81y2
Solution:
Using the product with a = 2x, b = 9y, we have
4x2 - 81y2 = ( 2x) 2 - ( 9y) 2
= ( 2x - 9y ) ( 2x + 9y ) |
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Example 4 |
x3y3 + 729
Solution:
Using the product with a = xy, b = 9, we have
x3y3 + 729 = (xy)3 + (9)3
= ( xy + 9 ) ( x2y2 - 9xy + 81 ) |
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Example 5 |
x3 - 6x2+ 12x - 8
Solution:
Using the product with a = x, b = 2, we have
x3 - 6x2+ 12x - 8 = ( x - 2 )3 |
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Example 6 |
x3 + y3 - z3+ 3xyz
Solution:
Using the product with a = x, b = y, c = -z, we have
x3 + y3 - z3+ 3xyz
= ( x + y - z ) ( x2+ y2+ z2 - xy + yz + zx ) |
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Example 7 |
n4+ 4
Solution:
n4+ 4 is in the form of a2 + b2, with a = n2, b = 2
n4+ 4 = a2+ b2
= ( a2+ b2+ 2ab ) - 2ab
= ( n4+ 4+ 2n2 (2)) - 2n2(2)
= ( n2+ 22- 4n)2
= (n2 + 22 - (2n)2
which is of the form a2- b2 with a = n2 + 2, b = 2n.
Therefore, ( n2 + 2) 2 - (2n2 = ( n2+ 2n + 2 ) ( n2 - 2n + 2 ).
Therefore, n4 + 4 = ( n2 + 2n + 2 ) ( n2 - 2n + 2 ). |
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Try these questions |
| 1. |
a2+ 4a + 4 |
2. |
a4+ 6a2+ 9a |
3. |
( 2x + 3y 2 + 2 ( 2x + 3y ) ( x + y ) + ( x + y 2 |
4. |
4a2 - 12ab + 9b2 |
5. |
9a2b2 - 6abc + c2 |
6. |
( 3x + 2y 2 - 2 ( 3x + 2y ) ( x + y ) + ( x + y) 2 |
7. |
4a2 - 9 |
8. |
16x2 - 9y2 |
9. |
a2b2 - c2d2 |
10. |
25a2 - 16a |
11. |
a4- 1 |
12. |
(a + b2 - c)2 |
13. |
a2 - ( b - c) 2 |
14. |
4a2 + 4ab + b2 - c2 |
15. |
a2 - b2 - c2 - 2bc |
16. |
a2 - b2 - 4c2 + 4bc |
17. |
x4 + 324 |
18. |
4a4 + 81 |
19. |
3x4 + 12 |
20. |
x4 + x2 + 1 |
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Answers |
| 1. |
a2 + 4a + 4 = (a2 + 2 (a) (2) + (22
= ( a + 2) 2 |
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2. |
a3+ 6a2+ 9a = a ( a2 + 6a + 9 )
= a [ (a2 + 2 (a) (3) + (3) 2 ]
= a [ a + 3 ]2 |
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3. |
( 2x + 3y 2 + 2 ( 2x + 3y ) ( x + y ) + ( x + y) 2
Put 2x + 3y = a; x + y = b
Therefore given expression = a2+ 2ab + b2
= ( a + b) 2
= ( 2x + 3y + x + y) 2
= ( 3x + 4y) 2 |
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4. |
4a2- 12ab + 9b2 = ( 2a )2 + 2 ( 2a ) ( - 3b ) + ( - 3b )2
= ( 2a - 3b )2 |
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5. |
9a2b2- 6abc + c2
= ( 3ab )2 + 2 ( 3ab ) ( - c ) + ( - c )2
= ( 3ab - c )2 |
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6. |
( 3x + 2y )2 - 2 ( 3x + 2y ) ( x + y ) + ( x + y )2
This is in the form of a2 - 2ab + b2
Therefore factors = ( a - b )2
i.e., ( 3x + 2y - x - y )2 = ( 2x + y )2 |
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7. |
4a2 - 9 = ( 2a )2 - (3)2
[Therefore a2 - b2 = ( a + b ) ( a - b ) ]
= ( 2a + 3 ) ( 2a - 3 ) |
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8. |
16x2- 9y2 = ( 4x )2 - ( 3y )2
= ( 4x + 3y ) ( 4x - 3y ) |
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9. |
a2b2 - c2d2= ( ab )2 - ( cd )2
= ( ab + cd ) ( ab - cd ) |
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10. |
25a2 - 16a = a ( 25a2 - 16 )
= a [ ( 5a )2- ( 4 )2]
= a ( 5a + 4 ) ( 5a - 4 ) |
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11. |
a4- 1 = ( a2 ) 2- (1)2
= ( a2 + 1 ) ( a2 - 1 )
= ( a2 + 1 ) ( a + 1 ) ( a - 1 ) |
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12. |
( a + b )2 - c2 = ( a + b + c ) ( a + b - c ) |
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13. |
a2- ( b - c )2 = ( a )2 - ( b - c )2
= ( a + b - c ) ( a - b + c ) |
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14. |
4a2 + 4ab + b2 - c2 = ( 2a )2 + 2 ( 2a ) ( b ) + ( b )2 - c2
= ( 2a + b )2 - ( c )2
= ( 2a + b + c ) ( 2a + b - c ) |
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15. |
a2 - b2 - c2- 2bc = a2 - ( b2 + c2 + 2bc )
= ( a )2 - ( b + c )2
= ( a + b + c ) ( a - b - c ) |
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16. |
a2 – b2 - 4c2 + 4bc = a2- ( b2+ 4c2 - 4bc )
= ( a )2- ( b - 2c )2
= ( a + b - 2c ) ( a - b + 2c ) |
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17. |
x4 + 324 = ( x )2 + (18)2 + 2 ( x )2 (18) - 36x2
= ( x2 + 18 )2 - ( 6x )2
= ( x2 + 18 + 6x ) ( x2 + 18 - 6x )
= ( x2 + 6x + 18 ) ( x2 - 6x + 18 ) |
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18. |
4a4 + 81 = ( 2a2 )2 + (9)2 + 2( 2a2 ) (9) -36a2
= ( 2a2 + 9 )2 - ( 6a )2
= ( 2a2 + 9 + 6a ) ( 2a2 + 9 + 6a )
= ( 2a2 - 6a + 9 ) ( 2a2 + 6a + 9 ) |
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19. |
3x4 + 12 = 3 [ x4 + 4 ]
= 3 [ ( x2 )2 + (2)2 + 2 ( x2 ) (2) - 4x2 ]
= 3 [ ( x2 + 2 )2 - ( 2x )2 ]
= 3 ( x2 + 2x + 2 ) ( x2 - 2x + 2 ) |
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20. |
x4 + x2 + 1 = ( x2 )2 + (1)2 + 2 ( x2 ) (1) - x2
= ( x2 + 1 )2 - (x)2
= ( x2 + x + 1 ) ( x2 - x + 1 ) |
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