Transcript Section 7.5 Factoring the Sum and Difference of Two cubes

Factoring the Sum and
Difference of Two cubes
3
a
3
b
+
a3 – b3
Count
1
1
1
• How long is the
edge?
• How many
squares in the
face?
• How many
blocks?
Count
Edge
Face Blocks
1
1
1
2
4
8
Count
Edge
Face Blocks
1
1
1
2
4
8
3
9
27
Count
Edge
n
1
Face Blocks
n2
n3
1
1
2
4
8
3
9
27
4
16
64
5
25
125
Memorize the First 10 Perfect Cubes
n
1
2
3
4
5
6
7
8
9
10
n2
1
4
9
16
25
36
49
64
81
100
n3
1
8
27
64
125
216
343
512
729
1000
Recall the Difference of Two
Squares Formula
–
=(a + b)(a – b)
2
x – 9 =(x + 3)(x – 3)
2
a
2
b
• There are similar formulas for the
sum and difference of two cubes.
Multiply a Binomial by a Trinomial
( x  xy  y )
2
2
The Sum of Cubes
x  y
3
(x  y)
 ( x  y )( x  xy  y )
2
x  x y  xy
3
2
x
3
2
 x y  xy  y
3
y
3
2
3
2
2
Difference of Cubes
( x  xy  y )
2
2
x y
3
(x  y)
 ( x  y )( x  xy  y )
2
x  x y  xy
3
2
x
3
2
 x y  xy  y
3
y
3
2
3
2
2
Compare the Formulas
The Sum of Cubes
x  y  ( x  y )( x  xy  y )
3
3
2
2
The Difference of Cubes
x  y  ( x  y )( x  xy  y )
3
3
2
They are just alike except for
where they are different.
2
Using the Difference of Cubes
x  y  ( x  y )( x  xy  y )
3
3
x
3
2
-8
Recall 23 = 8
= (x -
2
2)(x
+ 2x + 4)
2
Using the Sum of Cubes
x  y  ( x  y )( x  xy  y )
3
3
y
3
2
+ 27
Recall 33 = 27
= (y +
2
3)(y
– 3y + 9)
2
Factor Out the Common Factor
3xa + 2x + 21a + 14 =
3xa + 2x + 3(7)a + 2(7) =
x(3a + 2) + 7(3a + 2) = (3a + 2)(x +7)
This is called factoring by grouping.
What is factoring by grouping?
Factoring a common monomial from pairs of terms,
then looking for a common binomial factor is called
factor by grouping.
When do I use factoring by grouping?
*when the problem consists of 4 terms
How will my answer look?
*it will be the product of two binomials
Factor the expression
2
5 x ( x  2 )  3( x  2 )
2
5 x ( x  2)  3( x  2)
Pull the common factor out of each term.
2
( x  2) (5 x  3)
Notice what is left in each
term after factoring out the
common factor.
Notice
there are
two terms
Notice what
each term has
in common.
x2
Try this example:
7 y ( y  5)  3( y  5)
( y  5)(7 y  3)
Factor the polynomial
3
2
3
2
m  7 m  2 m  14
Form two
binomials with
a + sign
between them.
( m  7 m )  (  2 m  14)
2
m ( m  7 )  2 ( m  7)
2
( m  7 )( m  2 )
3
Try this example:
2
9x  9x  7x  7
3
2
(9 x  9 x )  (  7 x  7)
2
9 x ( x  1)  7( x  1)
2
( x  1)(9 x  7)
6x2 – 3x – 4x + 2 by grouping
– 3x – 4x + 2
2
= (6x – 3x) + (– 4x + 2)
= 3x(2x – 1) + -2(2x - 1)
= (2x – 1)(3x – 2)
2
6x
Homework
WB pp 89 and 90
Book
p. 78 #1-27 0dd,
p. 79 #1-27 odd
Page 78
Page 78