Factoring the Sum or Difference of Two Cubes The Sum of Cubes ...
Factoring the Sum or Difference of Two Cubes The Sum of Cubes ...
Factoring the Sum or Difference of Two Cubes The Sum of Cubes ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>Fact<strong>or</strong>ing</strong> <strong>the</strong> <strong>Sum</strong> <strong>or</strong> <strong>Difference</strong> <strong>of</strong> <strong>Two</strong> <strong>Cubes</strong><br />
<strong>The</strong> <strong>Sum</strong> <strong>of</strong> <strong>Cubes</strong> <strong>Fact<strong>or</strong>ing</strong> F<strong>or</strong>mula is:<br />
( )( )<br />
x + y = x + y x − xy + y<br />
3 3 2 2<br />
<strong>The</strong> <strong>Difference</strong> <strong>of</strong> <strong>Cubes</strong> <strong>Fact<strong>or</strong>ing</strong> F<strong>or</strong>mula is:<br />
( )( )<br />
x − y = x − y x + xy + y<br />
3 3 2 2<br />
Fact<strong>or</strong>:<br />
8a<br />
+ b<br />
3 3<br />
Fact<strong>or</strong>:<br />
8a<br />
− b<br />
3 3<br />
Fact<strong>or</strong>:<br />
8a<br />
+ 27b<br />
3 3<br />
Fact<strong>or</strong>:<br />
64x<br />
− 27y<br />
3 3<br />
Fact<strong>or</strong>:<br />
24a<br />
+ 3ax<br />
4 3
Answers<br />
Fact<strong>or</strong>:<br />
8a<br />
+ b<br />
3 3<br />
We begin by rewriting each term as a cube and use <strong>the</strong> sum <strong>of</strong> cubes f<strong>or</strong>mula as<br />
a guide.<br />
( )( )<br />
x + y = x + y x − xy + y<br />
3 3 2 2<br />
( )( )<br />
8 a + b = (2 a) + ( b) = 2 a+ b (2 a) − (2 a)( b) + ( b)<br />
3 3 3 3 2 2<br />
So simplifying <strong>the</strong> right hand side we get:<br />
( )( )<br />
3 3 2 2<br />
8a + b = 2a+ b 4a − 2ab+ b Answer<br />
Fact<strong>or</strong>:<br />
8a<br />
− b<br />
3 3<br />
We begin by rewriting each term as a cube and use <strong>the</strong> difference <strong>of</strong> cubes<br />
f<strong>or</strong>mula as a guide.<br />
( )( )<br />
x − y = x − y x + xy + y<br />
3 3 2 2<br />
( )( )<br />
8 a − b = (2 a) − ( b) = 2 a− b (2 a) + (2 a)( b) + ( b)<br />
3 3 3 3 2 2<br />
So simplifying <strong>the</strong> right hand side we get:<br />
( )( )<br />
3 3 2 2<br />
8a − b = 2a− b 4a + 2ab+ b Answer<br />
Fact<strong>or</strong>:<br />
8a<br />
+ 27b<br />
3 3<br />
We begin by rewriting each term as a cube and use <strong>the</strong> sum <strong>of</strong> cubes f<strong>or</strong>mula as<br />
a guide.<br />
( )( )<br />
x + y = x + y x − xy + y<br />
3 3 2 2<br />
( )( )<br />
8a + 27 b = (2 a) + (3 b) = 2a+ 3 b (2 a) − (2 a)(3 b) + (3 b)<br />
3 3 3 3 2 2<br />
So simplifying <strong>the</strong> right hand side we get:<br />
( )( )<br />
3 3 2 2<br />
8a + 27b = 2a+ 3b 4a − 6ab+ 9b<br />
Answer
Fact<strong>or</strong>:<br />
64x<br />
− 27y<br />
3 3<br />
We begin by rewriting each term as a cube and use <strong>the</strong> difference <strong>of</strong> cubes<br />
f<strong>or</strong>mula as a guide.<br />
( )( )<br />
x − y = x − y x + xy + y<br />
3 3 2 2<br />
( )( )<br />
64x − 27 y = (4 x) − (3 y) = 4x− 3 y (4 x) + (4 x)(3 y) + (3 y)<br />
3 3 3 3 2 2<br />
So simplifying <strong>the</strong> right hand side we get:<br />
( )( )<br />
3 3 2 2<br />
64x − 27y = 4x− 3y 16x + 12xy + 9y<br />
Answer<br />
Fact<strong>or</strong>:<br />
24a<br />
+ 3ax<br />
4 3<br />
This problem doesn’t even look like a sum <strong>or</strong> difference <strong>of</strong> cubes problem.<br />
However, <strong>the</strong>re is a common fact<strong>or</strong> in each <strong>of</strong> <strong>the</strong> two terms. It is 3a . So we<br />
can write:<br />
( )<br />
4 3 3 3<br />
24a + 3ax = 3a 8a + x Now notice that <strong>the</strong> fact<strong>or</strong> in paren<strong>the</strong>ses is a sum<br />
<strong>of</strong> two cubes and except f<strong>or</strong> <strong>the</strong> x it is <strong>the</strong> same expression that was fact<strong>or</strong>ed in<br />
<strong>the</strong> first exercise on this sheet. So we can write:<br />
( ) ( )( )<br />
4 3 3 3 2 2<br />
24a + 3ax = 3a 8a + x = 3a 2a+ x 4a − 2ax+ x Answer
Change language