Book 2A: Chapter 2, 4 – Identities and Factorization, More about Factorization of Polynomials
Revision
1.
Meaning of identities
(a) An equation that can be satisfied by ALL values of the unknown(s) is called
an identity.
(b) We use the symbol ‘ ≡ ’ instead of ‘=’ to represent an identity.
(c) The algebraic expressions on both sides of an identity are identical.
Therefore, we can find the unknown constant(s) in an identity by comparing
the like terms.
2.
Some important algebraic identities
a 2 + 2ab + b 2 ≡ (a + b) 2
a 2 − 2ab + b 2 ≡ (a − b)2
a2
− b 2 ≡ (a + b)(a − b)
a3
a3
3.
+ b3
− b3
≡ (a + b)(a 2 − ab + b 2 )
≡ (a − b)(a 2 + ab + b 2 )
Factorization
(a) The process of expressing an algebraic expression as a product of its factors
is called factorization.
(b) Factorization is the reverse process of expansion.
4.
Methods of factorization
(a) By taking out the common factor(s) from all the terms in the expression
until no more factors can be taken out.
(b) By grouping terms method
(c) By using identities
(d) By the Cross-method
1
Book 2A: Chapter 2, 4 – Identities and Factorization, More about Factorization of Polynomials
Example
1. Prove that x( x − 4) − 2( x − 1) ≡ x 2 − 2(3 x − 1) .
Solution:
L.H.S. = x( x − 4) − 2( x − 1)
= x2 − 4 x − 2 x + 2
= x2 − 6 x + 2
R.H.S. = x 2 − 2(3 x − 1)
2.
= x2 − 6 x + 2
i.e. L.H.S. = R.H.S.
therefore, x( x − 4) − 2( x − 1) ≡ x 2 − 2(3 x − 1)
Prove that −2( x − 3) = −2 x − 6 is not an identity.
Solution:
L.H.S. = −2( x − 3)
= −2 x + 6
R.H.S. = −2 x − 6
i.e. L.H.S. ≠ R.H.S.
therefore, −2( x − 3) = −2 x − 6 is not an identity.
3.
If 2(5 x + 2) ≡ Ax + B , where A and B are constants, find the values of A and B.
Solution:
L.H.S. = 2(5 x + 2) = 10 x + 4
i.e. 10 x + 4 = Ax + B
By comparing like terms, A = 10 and B = 4
4.
Expand the following expressions.
(a) ( x + 6)( x − 6) ;
(b) (2 x − 5)2
(c)
(7 − 3n)(−7 − 3n)
(e)
c d
 + 
2 3
(d) 3(−2 + a 2 )(−2 − a 2 )
2
t

w− 
3

(f)
2
Solution
(a) ( x + 6)( x − 6) = x 2 − 62 = x 2 − 36
(b) (2 x − 5) 2 = (2 x)2 − 2(2 x)(5) + (5) 2 = 4 x 2 − 20 x + 25
(c)
(7 − 3n)(−7 − 3n) = (−3n + 7)(−3n − 7) = (−3n) 2 − (7)2 = 9n 2 − 49
(d) 3(−2 + a 2 )(−2 − a 2 ) = 3[(−2) 2 − (a 2 ) 2 ] = 3(4 − a 4 ) = 12 − 3a 4
2
2
2
(e)
2
2
c d  c
 c  d   d  c cd d
+
=
+
2
+
=
+
+

  
    
4
3
9
2 3 2
 2  3   3 
(f)
t
t
t 2
2 wt t 2

2
2
w
−
=
(
w
)
−
2(
w
)(
)
+
(
)
=
w
−
+


3
3
3
3
9

2
2
Book 2A: Chapter 2, 4 – Identities and Factorization, More about Factorization of Polynomials
5.
Factorize the following expressions.
(a) 4 x + 6 xy
(b) 3 x + 3 y + bx + by
(c)
4(q − 1) − (1 − q )r
(d) ( x + y )2 (1 + x) − 3( x + y ) 2
(e)
3 p − 9q − 2 pq + 6q 2
(f)
ap + bp − cp + cq − bq − aq
(g) 3 x 2 − 48
(i) 9a 2 − 4b 2
(h) 2k 2 + 12k + 18
(j) 16(m − n) 2 − 1
(k) ( x + y )2 − (a + b)2
(l)
(m) 18 p 2 + 48 pq + 32q 2
(n) 4 x 2 + 12 x + 9 − y 2
(o) x − x 2 + 6
(q) 8 x3 + 1
(p) 16 − 12 x − 18 x 2
(r) 27 p 3 − 64q 3
−36a 2 + 12ab − b 2
Solution
(a) 4 x + 6 xy = 2 x(2) + 2 x(3 y ) = 2 x(2 + 3 y )
(b) 3 x + 3 y + bx + by = 3( x + y ) + b( x + y ) = ( x + y )(3 + b)
(c)
4(q − 1) − (1 − q )r = 4(q − 1) + (q − 1)r = (q − 1)(4 + r )
(d) ( x + y ) 2 (1 + x) − 3( x + y )2 = ( x + y ) 2 (1 + x − 3) = ( x + y ) 2 ( x − 2)
(e)
(f)
3 p − 9q − 2 pq + 6q 2 = 3( p − 3q ) − 2q ( p − 3q ) = ( p − 3q )(3 − 2q )
ap + bp − cp + cq − bq − aq
= p ( a + b − c ) + q (c − b − a )
= p(a + b − c) − q(a + b − c)
= (a + b − c)( p − q )
(g) 3 x 2 − 48 = 3( x 2 − 16) = 3( x 2 − 4 2 ) = 3( x + 4)( x − 4)
(h) 2k 2 + 12k + 18 = 2(k 2 + 6k + 9) = 2[(k ) 2 + 2(k )(3) + (3) 2 ] = 2(k + 3) 2
(i)
9a 2 − 4b 2 = (3a ) 2 − (2b)2 = (3a + 2b)(3a − 2b)
(j)
16(m − n) 2 − 1
= [4(m − n)]2 − (1) 2
= [4(m − n) + 1][4(m − n) − 1]
= (4m − 4n + 1)(4m − 4n − 1)
(k) ( x + y )2 − (a + b)2
= [( x + y ) + (a + b)][( x + y ) − (a + b)]
= ( x + y + a + b)( x + y − a − b)
(l)
−36a 2 + 12ab − b 2
= −(36a 2 − 12ab + b 2 )
= −[(6a ) 2 − 2(6a )(b) + (b) 2 ]
= −(6a − b)2
(m) 18 p 2 + 48 pq + 32q 2
3
Book 2A: Chapter 2, 4 – Identities and Factorization, More about Factorization of Polynomials
= 2(9 p 2 + 24 pq + 16q 2 )
= 2[(3 p ) 2 + 2(3 p )(4q ) + (4q )2 ]
= 2(3 p + 4q ) 2
(n) 4 x 2 + 12 x + 9 − y 2
= (2 x)2 + 2(2 x)(3) + (3)2 − y 2
= (2 x + 3)2 − y 2
= (2 x + 3 + y )(2 x + 3 − y )
(o)
x − x 2 + 6 = − x 2 + x + 6 = −( x 2 − x − 6) = −( x − 3)( x + 2)
(p) 16 − 12 x − 18 x 2 = −18 x 2 − 12 x + 16 = −2(9 x 2 + 6 x − 8) = −2(3 x + 4)(3 x − 2)
(q) 8 x 3 + 1 = (2 x)3 + (1)3 = (2 x + 1)[(2 x) 2 − (2 x)(1) + (1) 2 ] = (2 x + 1)(4 x 2 − 2 x + 1)
(r)
27 p 3 − 64q 3
= (3 p )3 − (4q )3
= (3 p − 4q )[(3 p ) 2 + (3 p )(4q ) + (4q )2 ]
= (3 p − 4q )(9 p 2 + 12 pq + 16q 2 )
4
Book 2A: Chapter 2, 4 – Identities and Factorization, More about Factorization of Polynomials
Exercise
1
Simplify the following expressions.
(a) (2a – b) + (3a + 2b) – (4a – 3b)
2
(b) (3c + 2d) – (d – 2c) + (3d + 2c)
Simplify the following expressions.
(a) (3x – y)(x + 2y)
3
(b) (2y + 3x)(–x + y)
Factorize the following expressions.
(b) 2a2 – 2a + b – ab
(a) 4c – 6d + 8
4
Identify whether each of the following equations is an identity.
(a) 6a(a – 2b) = 6a2 – 12ab
5
Factorize the following expressions.
(a) 4a2 – 1
6
(b) 9x2 – 12xy + 4y2
(b) m2 – 9n2
(b) b2 + 18b + 81
(b) 4b2 – 20b + 25
(c) 9c2 – 12c + 4
(b) 2b2 + b – 1
Factorize the following polynomials.
(a) 2c2 – 3c – 2
12
(c) c2 – 22c + 121
Factorize the following polynomials.
(a) a2 – a – 2
11
(c) 64r2 – 25s2
Factorize the following polynomials.
(a) 25a2 – 30a + 9
10
(c) 1 – 16c2
Factorize the following polynomials.
(a) a2 – 12a + 36
9
(b) b2 – 100
Factorize the following expressions.
(a) 4b2 – 49
8
(c) m2 + 10mn + 25n2 (d) 2x2 – 8y2
Factorize the following polynomials.
(a) 25 – a2
7
(b) 2xy – 2x + y – 1 = (2x – 1)(y + 1)
(b) 3x2 + x – 2
Factorize the following polynomials.
(a) 1 – 8a3
(b) b3 + 64
(c) 1 + (mn)3
5
Book 2A: Chapter 2, 4 – Identities and Factorization, More about Factorization of Polynomials
13
Factorize the following expressions.
(a) –36a2 + b2
14
Factorize the following expressions.
(a) (3 – m)2 – 144
15
(b) 81a4 – 256
Factorize the following expressions.
(a) 50a2 – 20a2b2 + 2b2
24
(b) (2x – y)2 – (x + 2y)2
Factorize the following expressions.
(a) 25m2 – 36n2 – 5m – 6n
23
(b) 192a3 – 375b3
Factorize the following expressions.
(a) (a – b)2 – (a + b)2
22
(b) m3n3 – 216p3
Factorize the following expressions.
(a) 16x3 + 2y3
21
(b) –4a2 – 10ab + 6b2
Factorize the following expressions.
(a) 8x3 – 125y3
20
(b) 14b2 – 13b + 3
Factorize the following expressions.
(a) 6a2 + 13ab – 5b2
19
(b) 3a2 + 12ab + 12b2
Factorize the following polynomials.
(a) 6a2 – a – 2
18
(b) 16x2 + 56xy + 49y2
Factorize the following expressions.
(a) 8m2 – 8mn + 2n2
17
(b) (x + 2y)2 – 9y2
Factorize the following expressions.
(a) 9a2 – 30ab + 25b2
16
(b) 4x2y2 – 49z2
(b) 16(x + y)2 – 24x(x + y) + 9x2
(a) Factorize 16a2 – 16ab + 4b2.
(b) Use the result in (a) to factorize 16a2 – 16ab + 4b2 – 25c2.
25
Factorize the following expressions.
(a) 60a2 – 28ab – 8b2
(b) –42c2 + 68cd – 16d2
6
Book 2A: Chapter 2, 4 – Identities and Factorization, More about Factorization of Polynomials
26
(a) Factorize 9m2 – 6mn + n2.
(b) Use the result in (a) to factorize 9m2 – 6mn + n2 + 6mp – 2np + p2.
27
Factorize the following expressions.
(a) (2x – y)3 – (x + y)3
28
(b) 512a3 – 8b3
(a) Factorize 27m3 – 1.
(b) Use the result in (a) to factorize 27m3 + 9m – 4.
29. Factorize 3x + 6y – 12z.
30. Factorize −12pqr – 15qr −9pq.
31. Factorize 5ef 2 −10de2 – 10e3.
32. Factorize 12x2y + 18xy2 + 6x3y2.
33. Factorize m(k − 3) + n(k − 3) .
34. Factorize − 1 + x − w(1 − x) .
35. Factorize (4k − 1)( x + y ) − (2k − 5)( y + x) .
36. Factorize 7(m − n) 3 − 49(n − m) 2 .
37. (a) Factorize 2m – 6n and 9n – 3m.
(b) Hence, factorize 6m(2m – 6n) – 15n(9n – 3m) + m – 3n.
38. Factorize 2y3 + 10y2 – 4y − 20.
39. Factorize r2 − 4rs + 12a2s – 3a2r.
40. Factorize 3a2 – 10bd + 5ab – 6ad.
41. Factorize a(p – 2)2 + 14b – 7bp.
42. Factorize 4ax2z – 12a2z + 8ax2y – 24a2y.
7
Book 2A: Chapter 2, 4 – Identities and Factorization, More about Factorization of Polynomials
43. Factorize as + bs – cs + ct – bt − at.
44. Factorize mx – 2my – px + 2py + 3mz − 3pz.
45. Factorize (3p + q)2 – 4.
46. Factorize −9(r + 1)2 + 36r2.
47. Factorize a2 + (2 + a)2 – (2 – a)2.
48. Factorize p2 – q2 + 5p – 5q.
49. Factorize 64x2 – 8x – 25y2 – 5y.
50. Factorize 1 – 16x4.
51. Factorize 32a2 – b2 – 2a2b2 +16.
52. (a) Factorize y3 – yx2 and x3 + x2y.
(b) Hence, factorize x3 + y3.
53. Factorize 27bx2 + 18bxy + 3by2.
54. Factorize –r5 – 18r3 – 81r.
55. Factorize (y – 2)2 – 24(y – 2) + 144.
56. Factorize 36(x – y)2 + 12z(x – y) + z2.
57. Factorize x2 + 10x + 25 – 4y2.
58. Factorize 4 x 2 − 20 xy + 25 y 2 + 10 y − 4 x.
59. (a) Factorize 4a2 – 28ab + 49b2.
(b) Hence, factorize 4(x – 1)2 – 28(x – 1)(y + 5) + 49(y + 5)2.
60. (a) Factorize 5x2 + 20x + 20.
(b) Hence, factorize 5x2 + 20x − 25.
8