Download HFCC Math Lab Beginning Algebra - 17

HFCC Math Lab
Beginning Algebra – 17
FACTORING THE DIFFERENCE OF TWO SQUARES
Introductory Terminology
Terms
Definitions
Product
An indicated multiplication; for example: 2 ⋅ 3, 4 ⋅ a, 7b,
5t(a + 3) are all products.
Factor
Any one of the numbers or expressions being multiplied to
form a product; for example: in 4 ⋅ a, 4 and a are factors;
in 5t(a + 3), 5, t, (a + 3) are factors.
Perfect square
Any number or expression that can be obtained by
multiplying a quantity by itself; for example: 25 is a perfect
square since 25 = 5 ⋅ 5 or 52 .
4t 2 is a perfect square since 4t 2 = (2t)(2t) = (2t )2 .
Objective I.
To identify perfect squares
Objective II.
To write the difference of two squares as the product of two
factors.
I. EXAMPLES:
Directions:
1.
9x 2
Identify which of the following are perfect squares.
Think: What value times itself equals 9x 2 ?
(3 x) 2 = 9x 2 , so 9x 2 is a perfect square.
2.
64
Think: What value times itself equals 64? 82 = 64,
so 64 is a perfect square.
3.
y4
Think: What value times itself equals y 4 ?
y 4 = y 2 ⋅ y 2 = ( y 2 )2 . y 4 is a perfect square.
4.
y5
Revised 03/09
Think: The power of a perfect square is an even number.
y 5 is not a perfect square since it is impossible to
square a whole number power and get an odd power.
1
5.
y6
y 6 is a perfect square since y 6 = ( y 3 ) 2 .
6.
y9
y 9 is not a perfect square by the same reason as #4.
7.
1
1 is a perfect square since 12 = 1.
8.
x2 y 4
x 2 y 4 is a perfect square since x 2 y 4 = ( xy 2 ) .
2
II.
To factor (write as a product) any expression of the form ( A) − ( B ) , re-write
this as the product of a sum times a difference, namely (A + B)(A – B);
that is, A2 − B 2 = ( A + B )( A − B ) or ( A − B)( A + B) .
2
2
EXAMPLES: Directions: Factor each expression completely.
1.
64 − t 2
Think: Is 64 a perfect square? Yes, 64 = 82 .
Is t 2 a perfect square? Yes, t 2 = (t ) 2 .
64 − t 2 = (8)2 − (t ) 2 . To factor a difference of perfect
squares, use the formula above. 64 − t 2 = (8 + t)(8 – t).
2.
9x2 −1
Think: Is 9 x 2 a perfect square? Yes, 9 x 2 = (3 x) 2 .
Is 1 a perfect square? Yes, 1 = (1) 2 .
9 x 2 − 1 = (3 x) 2 − (1)2 . To factor a difference of perfect
squares, use the formula above. 9 x 2 − 1 = (3x + 1)(3x – 1).
3.
64 + t 2
Think: This is not a difference of two squares and thus cannot be
factored as above.
4.
y 4 − 25
Can you think this through for yourself?
y 4 − 25 = ( y 2 )2 − (5) 2 , so y 4 − 25 = ( y 2 − 5)( y 2 + 5) .
5.
x 2 y 4 − 9t 2
Try this for yourself, too. The answer is:
x 2 y 4 − 9t 2 = ( xy 2 ) 2 − (3t )2 and so equals the product
( xy 2 − 3t )( xy 2 + 3t ) .
6.
5 x 2 − 45
Think: Are 5 x 2 or 45 perfect squares?
Can you factor a common factor from 5 x 2 and 45?
Revised 03/09
2
The expression 5 x 2 − 45 , has 5 as a common factor and can be
factored as 5( x 2 − 9), now factor x 2 − 9 .
So, 5 x 2 − 45 = 5(x - 3)(x + 3).
Exercises: Factor completely.
1.
p2 − q 2
11.
b 2 c 2 − 9d 2
2.
y2 − 9
12.
25m2 − 16n 2
3.
9 −t2
13.
9 g 2 − 121
4.
4a 2 − 9b 2
14.
4 x 2 − 16 y 2
5.
16b 2 − 1
15.
81 − 4h 2
6.
25 y 2 − 9
16.
x 4 − 16
7.
a 2 − 36b 2
17.
a4 − 1
8.
49 x 2 − 16 y 2
18.
2t 2 − 32
9.
4 − 9x2
19.
28a 3 − 7at 2
10.
100 − x 2 y 2
20.
3a 2b 2 − 12c 2
Solutions to odd-numbered problems and answers to even-numbered problems:
1.
( p 2 − q2 )
(p + q)(p – q)
2.
(y – 3)(y + 3)
3.
9 −t2
32 − t 2
(3 – t)(3 + t)
14.
4(x -2y)(x + 2y)
15.
81 − 4h 2
92 − (2h)2
(9 – 2h)(9 + 2h)
(2a + 3b)(2a – 3b)
16.
( x − 2)( x + 2)( x 2 + 4)
4.
Revised 03/09
13.
9 g 2 − 121
(3 g )2 − 112
(3g – 11)(3g + 11)
3
5.
16b 2 − 1
(4b)2 − 12
(4b+1)(4b-1)
17.
(a 2 − 1)(a 2 + 1)
(a − 1)(a + 1)(a 2 + 1)
6.
(5y – 3)(5y + 3)
18.
2(t – 4)(t + 4)
19.
28a 3 − 7at 2
7a (4a 2 − t 2 )
7.
a 2 − 36b 2
2
( a ) − (6b)2
(a + 6b)(a – 6b)
2
7 a  ( 2a ) − t 2 


8.
(7x + 4y)(7x – 4y)
7a(2a – t)(2a + t)
9.
4 − 9x 2
22 − (3 x)2
(2 – 3x)(2 + 3x)
10.
(10 – xy)(10 + xy)
11.
(bc )2 − (3d ) 2
(bc + 3d)(bc – 3d)
12.
(5m – 4n)(5m + 4n)
20.
3(ab - 2c)(ab + 2c)
NOTE: You can get additional instruction and practice by going to the
following websites.
http://regentsprep.org/Regents/math/factor/Lfactps.htm This website
provides step by step instruction and exercises.
http://algebralab.org/practice/practice.aspx?file=Algebra_Redden4B.x
ml This website provides one worksheet with answers.
http://www.cbv.ns.ca/gbhigh/pauloneill/Objective17.08.pdf This website
provides step by step instruction and exercises.
Revised 03/09
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