F.2 Mathematics Supplementary Notes
Chapter 3 Factorization of Simple Polynomials
Chapter 3 Factorization of Simple Polynomials
Important Terms
Factorization
Common Factor
Polynomial
因式分解
公因式
多項式
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Name:_________( ) Class: F.2(
)
恆等式
併項
展開
Identity
Grouping Terms
Expansion
Revision Notes:
1.
More about Factorization of Polynomials
(a) 5x and 2x2 + 1 are the factors of the polynomial 10x3 + 5x.
5x is the common factor of all the terms of 10x3 + 5x.
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 10x3 + 5x = 5x(2x2 +1)
(b) To factorize a polynomial is to express the polynomial as a product of its factors.
Example 1. 10x3 + 5x = 5x(2x2 +1 )
(c) Factorization of polynomials is the reverse process of expansion.
(d) There are different methods of factorization. The commonly used methods are: taking out
common factors, grouping terms, using identities and the cross-method.
2.
Taking out Factors and Grouping Terms
(a) Taking out common factors
Example 2. –5a – 5b = –5(a + b)
(b) Grouping terms
Example 3.
cd – c + d2 – d = c(d – 1) + d(d – 1)
= (d – 1)(c + d)
3.
Using Identities
(a) The following identities can be used to factorize the polynomials in the form of a2 – b2,
a2 + 2ab + b2 or a2 – 2ab + b2.
a2 – b2  (a + b)(a – b)
a2 + 2ab + b2  (a + b)2
a2 – 2ab + b2  (a – b)2
Example 4.
Factorize (i)
9 x 2  16 y 2 .
(ii) 16a 4  36
(iii) (4  ab 2 ) 2  4a 2 b 4
F.2 Mathematics Supplementary Notes
Solution (i)
Chapter 3 Factorization of Simple Polynomials
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9 x 2  16 y 2  (3x) 2  (4 y) 2
 (3x  4 y )(3x  4 y )
(ii) 16a 4  36  4(4a 4  9)
 4[( 2a 2 ) 2  (3) 2 ]
= 4(2a 2  3)(2a 2  3)
(iii) (4  ab 2 ) 2  4a 2 b 4  (4  ab 2 ) 2  (2ab 2 ) 2
 (4  ab 2  2ab 2 )( 4  ab 2  2ab 2 )
 (4  3ab 2 )( 4  ab 2 )
Example 5.
Factorize(i) x 2  8 x  16
(ii) 25x 2  30 xy  9 y 2
Solution (i)
x 2  8 x  16  ( x) 2  2(4)( x)  (4) 2
 ( x  4) 2
(ii) 25 x 2  30 xy  9 y 2  (5x) 2  2(5x)(3 y)  (3 y) 2
 (5 x  3 y ) 2
(b) The following identities can be used to factorize the polynomials in the form of
a3 – b3 or a3 + b3:
a3 – b3  (a – b)( a2 + ab + b2)
a3 + b3  (a + b)( a2 – ab + b2)
Example 6.
4.
Factorize 135  40a 3 .
Solution 135  40a 3 =
=
=
=
5(27  8a 3 )
5[(3) 3  (2a) 2 ]
5(3  2a)[(3) 2  (3)( 2a)  (2a) 2 ]
5(3  2a)(9  6a  4a 2 )
Cross-method
This method can be used to factorize quadratic polynomials of the form px2 + qx + r.
e.g. Factorize x2 – 4x + 3.
Since the x2 term can be written as (x) (x), and the constant term + 3 can be written as
(+1) (+3) or (–1) (–3), we can list out all the possible pairs of factors as follows:
x
x
+1 –1
+3 –3
Using the cross-method to test each possible pair of factors, we have x2 – 4x +3 = (x –1)(x – 3)
F.2 Mathematics Supplementary Notes
Chapter 3 Factorization of Simple Polynomials
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5. Harder Examples
Example 7.
Factorize a 2  2ab  b 2  6a  6b  9
a 2  2ab  b 2  6a  6b  9
Solution
 (a 2  2ab  b 2 )  6(a  b)  9
 (a  b) 2  2(3)(a  b)  (3) 2
 [(a  b)  3]2
 (a  b  3) 2
Example 8.
Factorize (i)
(ii)
Solution (i)
(ii)
ma  mb  na  nb
ax  bx  by  cy  cx  ay
ma  mb  na  nb
 (ma  mb)  (na  nb)
 m( a  b )  n ( a  b )
 (a  b)( m  n)
ax  bx  by  cy  cx  ay
 ax  bx  cx  ay  by  cy
 (ax  bx  cx)  (ay  by  cy )
Method II
ax  bx  by  cy  cx  ay
 ax  ay  bx  by  cx  cy
 (ax  ay )  (bx  by )  (cx  cy )
 a ( x  y )  b( x  y )  c ( x  y )
 ( x  y )( a  b  c)
 x(a  b  c)  y (a  b  c)
 (a  b  c)( x  y )
Example 9.
Factorize (i)
1
27 m 2  n 2
3
(ii) x4 + 4
Solution (i)
1
1
27 m 2  n 2 = (81m 2  n 2 )
3
3
=
(ii) x 4  4
1
(9m  n)(9m  n)
3
= x 4  4x 2  4  4x 2
= ( x 2  2) 2  (2 x) 2
= ( x 2  2 x  2)( x 2  2 x  2)
F.2 Mathematics Supplementary Notes
Chapter 3 Factorization of Simple Polynomials
Exercise A
Level I
Factorize:
1. (a) –7c – 7d
2.
(b)
–12g2 + 4g
(c) –a2bc – acd
(d)
9a2 + 18a3b – 15a2b2
(e) 4(x – 2) – (x – 2)(x + 1)
(f)
(a + 2)2 – (a + 2)
(a) 2u3 + u2 + 8u + 4
(b)
st – 3s – 2t + 6
(c) 1 + x – y – xy
(d)
xy – ay + 2x – 2a
(b)
m( a  b  c )  n (  a  b  c )
Level 2
Factorize:
3.
(a)
 p  q 2  p  q
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F.2 Mathematics Supplementary Notes
4.
Chapter 3 Factorization of Simple Polynomials
(c) ab(x – 5) – b(5 – x)
(d)
(a – b)(2a +b) + (b – a)(3a – 2b)
(e) (a – b)2 – 2c(b – a)
(f)
(4b – c)(x2 + 2) + (x2 – 2)(c – 4b)
(a) ab(x2 + y2) + xy(a2 + b2)
(b)
a(b + c + d) – d(a + b + c)
(b)
49b2 – 1
(c) 25y2 – 16
(d)
8x2 – 2
(a) a2 – 2a + 1
(b)
b2 + 12b + 36
Exercise B
Level 1
Factorize:
1. (a) y2 – 1
2.
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F.2 Mathematics Supplementary Notes
(c) 8m2 – 8m + 2
Chapter 3 Factorization of Simple Polynomials
(d)
4 – 12n + 9n2
(b)
343 + y3
(c) 216 + a3b3
(d)
250a3 – 16
(a) a4 – 625
(b)
a 8  16
(c) 4x2 – 12x + 9 – y2
(d)
9a2 + 24ab + 16b2
Level 2
Factorize:
3. (a) a3 – 64
4.
2
1 
1

(d)  x     x  
x 
x

2
(e)
144  x 2  8xy  16 y 2
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F.2 Mathematics Supplementary Notes
Chapter 3 Factorization of Simple Polynomials
Exercise C
Level 1
Factorize:
1. (a) x2 + 11x + 18
2.
(b)
x2 – 12x + 11
(c) m2 – 3m + 2
(d)
b2 + 5x – 14
(a) 2b2 + 7b + 6
(b)
3t2 – 4t – 7
(c) 5a2 + 14ab – 3b2
(d)
3p2 – 4pq – 7q2
Level 2
3. (a) Factorize 3x 2  5xy  2 y 2 .
(b) Hence factorize 3( x  1) 2  5( x  1)( y  1)  2( y  1) 2 .
4.
(a) Factorize 4 x 2  12 x  9 .
(b) Hence factorize xy 2  12 x 2  4 x 3  9 x .
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F.2 Mathematics Supplementary Notes
Chapter 3 Factorization of Simple Polynomials
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-------------------------------------------------Optional------------------------------------------------Level III
1. Factorize
(a) x4 + xy + y4
2.
(b)
y 4  15 y 2  49
Factorize
(b) Factorize 3n 3 
(a) 2w 6  8w 3  2
1
9
(a) Prove that a 3  b 3  c 3  3abc  (a  b  c)(a 2  b 2  c 2  ab  bc  ca) .
(b) Hence factorize x 3  8 y 3  z 3  6 xyz .
(Ans: ( x  2 y  z )( x 2  4 y 2  z 2  2 xy  xz  2 yz ) )
3.
*4. Factorize bc(b  c)  ca(c  a)  ab(a  b)
(Ans: (a+b)(b+c)(c–a))
*5. Given that a  b  c   3(a 2  b 2  c 2 ) and a+b+c =12 , find the value of a.
2
(Ans: a = 4)
*6. Factorize (a) x 4  2 x 3  3x 2  2 x  1
(b) ( x  1)( x  2)( x  3)( x  4)  120
(Ans: (a) ( x 2  x  1) 2
(b) ( x 2  5 x  16)( x  6)( x  1))
(01HKMO)
F.2 Mathematics Supplementary Notes
Chapter 3 Factorization of Simple Polynomials
數學課外閱讀：＜＜數學誕生的故事＞＞。袁小明編著。九章出版社。
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Chapter 3 Factorization of Simple Polynomials
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Chapter 3 Factorization of Simple Polynomials
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Chapter 3 Factorization of Simple Polynomials
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Chapter 3 Factorization of Simple Polynomials
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