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In this presentation we will be studying special factoring formulas for:A) Perfect Square Trinomials B) The difference of two squaresC) The sum or difference of two cubes.
A) Perfect Square Trinomials
Perhaps you remember the formula for squaring a binomial:
The right side of each of the above equations is called a “perfect square trinomial.” You can factor a perfect square trinomial by trial and error but, if you recognize the form, you can do the job easier and faster by using the factoring formula.
222
222
bab2aba
bab2aba
)(
)(
To factor a perfect square trinomial you look for certain clues:1. Is the first term a perfect square of the forma2 for some expression a?
2. Is the last term a perfect square of the formb2 for some expression b?
3. Is the middle term plus or minus 2 times the square root of a2 times the square root of b2?
1x2x2
For example: Is the following a perfect square trinomial?
1x2x2
Is this term a perfect square? Yes.
Is this term a perfect square? Yes.
Is the middle term plus or minus 2 times the square root of x2 times the square root of 12?
x times 1 times 2 = 2x yes.
Then this is a perfect square and factors into(x + 1)2
See if you can recognize perfect square trinomials.
1. 25x10x2 Yes, this is (x + 5)2
2. 4x4x2 Yes, this is (x + 2)2
Yes, this is (3y - 2)23. 4y12y9 2
y122y32
24y3y9 2
...
36y12y2 4.
No, -36 is not a perfect square
T h i s p o i n t c o n c e r n i n g t h e m i d d l e t e r m i s i m p o r t a n t . C o n s i d e r t h e f o l l o w i n g e x a m p l e :
F a c t o r .91 44 2 xx T h e fi r s t t e r m a n d t h e l a s t t e r m a r e b o t h p e r f e c t s q u a r e s b u t t h e m i d d l e t e r m i s n o t t r u e t o f o r m . 2 ( 2 ) ( 3 ) = 1 2 o r – 1 2 s h o u l d b e t h e m i d d l e t e r m .
B y t r i a l a n d e r r o r .9149144 2 xxxx W e c o u l d h a v e a p e r f e c t s q u a r e t r i n o m i a l b y c h a n g i n g t h e m i d d l e t e r m :
22 3232329124 xxxxx .
Is the middle term 2 times the square root of a2 times the square root of b2
to Rameshwaram is mandatory. Rameshwaram is also popularly referred to as the 'Benaras of the south'. Rameshwaram is significant to Rameshwaram is mandatory. Rameshwaram is also popularly referred to as the 'Benaras of the south'. Rameshwaram is significant
Try the following four problems. Watch for thepattern. Do all four before you check theanswers on the next slide.
49284
64489
3613
6416
2
2
2
2
xx
xx
xx
xx
T h e s o l u t i o n s a r e :
22
22
2
22
72727249284
83838364489
493613
8886416
xxxxx
xxxxx
xxxx
xxxxx
T h e s e c o n d p r o b l e m w a s n o t a p e r f e c t s q u a r eb u t w a s f a c t o r e d a s a s i m p l e t r i n o m i a l . T h eo t h e r s w e r e p e r f e c t s q u a r e t r i n o m i a l s .
Try four more. Watch for the pattern for perfect square trinomials: the first term and the last term are perfect squares and the middle term is plus or minus 2ab. If your trinomial is not a perfect square, proceed by trial and error.
1610
169
254016
10020
2
2
2
2
xx
xx
xx
xx
Do all four before you go to the next slide.
H e r e a r e t h e s o l u t i o n s .
)10(21610
131313169
545454254016
10101010020
2
22
22
22
xxxx
xxxxx
xxxxx
xxxxx
T h e f i r s t t h r e e t r i n o m i a l s w e r e p e r f e c t s q u a r e s . N o t e t h a t i n e a c h c a s e , t w i c e t h e p r o d u c t o f t h e t e r m s i n t h e a n s w e r i s e x a c t l y t h e m i d d l e t e r m o f t h e t r i n o m i a l . T h e l a s t t r i n o m i a l i s n o t a p e r f e c t s q u a r e t r i n o m i a l .
B) The Difference of Two Squares
In the difference of two squares
This side is expanded
yxyxyx 22 This side is factored.
x (in the factored side) is the square root of x2 and y (in the factored side) is the square root of y2
Factoring the difference of two squares: some examples.
Note: The sum of two squares is not factorable.
36x2 Why does this binomial not factor?
Suppose it did. What would be the possible factors?
36x12x6x6x 2
3666
3612662
2
xxx
xxxx
No
No
There are no other possibilities, thus it doesn’t factor. In general, the sum of two squares does not factor.
No
Factoring the difference of two squares: Write out your answers before you go to the next page.
The solutions are:
1717149
98988164
5353259
1010100
2
2
2
2
xxx
xxx
yyy
xxx
Factoring the difference of two squares: Write down your solutions before going to the next page.
Factoring the difference of two squares: Here are the solutions.
C) The sum or difference of two cubes
F i n a l l y , w e l o o k a t t w o s p e c i a l p r o d u c t s i n v o l v i n g c u b e s . T h e s u m o f t w o c u b e s : 2233 babababa T h e d i f f e r e n c e o f t w o c u b e s : 2233 babababa
1000 is a perfect cube since 1000 = 103
125 is a perfect cube since 125 = 53
64 is a perfect cube since 64 = 43
8 is a perfect cube since 8 = 23
1 is a perfect cube since 1 = 13
In order to use these two formulas, you must be able to recognize numbers that are perfect cubes.
The following can be factored as the difference of two cubes: letting a = 2x and b = 3
278 3 x
2233 babababa
9643232 233 xxxx
Let’s check with multiplication to see if the factors are correct:
278
27181218128964323
2232
x
xxxxxxxx
F o r a n o t h e r e x a m p l e , t h e e x p r e s s i o n 643 x c a n b ef a c t o r e d a s t h e s u m o f t w o c u b e s u s i n g
))(( 2233 babababa w i t h xa a n d4b .
164464 23 xxxx
T o c h e c k t h i s a n s w e r w i t h m u l t i p l i c a t i o n ,
.64
6416416416443
2232
x
xxxxxxxx
A way to remember the formula
a3 + b3 = ( )( )a + bCube root of a and b with same sign
Clues to remember the formula:1. Cubes always factor into a binomial times a trinomial.
2. The binomial in the factored version always contains the cube roots of the original expression with the same sign that was used in the original expression.
a3 - b3 = ( )( )a - bCube root of a and b with same sign
A way to remember the formula
a3 + b3 = ( )( )a + b a2
1)Square first term
-ab
2)Find the product of both terms and change the sign i.e. a(b) =ab change the sign = -ab
+b2
3)Square last term i.e. last term is b2
Next you use the binomial to build your trinomial:
a3 - b3 = ( )( )A binomial times a trinomial
a - bCube root of a and b with same sign
a2
1)Square first term
+ab
2)Find the product of both terms and change the sign
+b2
3)Square last term
The difference of two cubes:
Cube Numbers you will findin problems.
13=123=8
33=27
43=64
53=125 etc3=etc
Difference of two cubes
8x3 - 125= ( )( )A binomial times a trinomial
2x - 5Cube root of 1st and 2nd term with same sign
4x2
1)Square first term
+10x
2)Find the product of both terms and change the sign
+25
3)Square last term
Build trinomialwith binomial.
Sum or Differenceof two cubes
64x3 + 1= ( )( )A binomial times a trinomial
4x + 1 16x2 -4x + 1
Build trinomial with binomial.
Now try the following four problems on yourown. The answers are on the next page.Factor each of the following as the sum oftwo cubes or as the difference of two cubes.
1000
64
1258
27
3
3
3
3
x
y
x
x
Here are the solutions to the four factoring problems with the formula substitutions for a and b at the end.
10,
10010101000
,4
416464
5,2
25104521258
3,
93327
23
23
23
23
bxa
xxxx
yba
yyyy
bxa
xxxx
bxa
xxxx
Now try four more problems on your own.The answers are on the next page. Factoreach of the following as the sum of two cubesor as the difference of two cubes.
3
3
3
3
12527
164
125
2764
x
x
x
x
H e r e a r e t h e s o l u t i o n s t o t h e f o u r f a c t o r i n gp r o b l e m s o n t h e p r e v i o u s p a g e .
23
23
23
23
251595312527
141614164
2555125
91216342764
xxxx
xxxx
xxxx
xxxx
T his is t he end of t he unit on special f act or ing. A t t his point you should move on t o t he pract ice t est f or special f act or ing. For t he exam, you will need t o memor ize the f ollowing f ormulas:
2233
2233
22
222
222
2
2
babababa
babababa
bababa
bababa
bababa
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