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# Factoring Cubes

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### Factoring Cubes

1. 1. Factoring the Sum & Difference of Two Cubes p.368-371
2. 2. This is a piece of cake, if you have perfect cubes. What are perfect cubes?
3. 3. This is a piece of cake, if you have perfect cubes. What are perfect cubes? Something times something times something. Where the something is a factor 3 times. 8 is 2 × 2 × 2, so 8 is a perfect cube. x6 is x2 × x2 × x2 so x6 is a perfect cube. It is easy to see if a variable is a perfect cube, how?
4. 4. This is a piece of cake, if you have perfect cubes. What are perfect cubes? Something times something times something. Where the something is a factor 3 times. 8 is 2 × 2 × 2, so 8 is a perfect cube. x6 is x2 × x2 × x2 so x6 is a perfect cube. It is easy to see if a variable is a perfect cube, how? See if the exponent is divisible by 3. It’s harder for integers.
5. 5. The sum or difference of two cubes will factor into a binomial × trinomial. ( a + b = ( a + b ) a − ab + b 3 3 2 2 ) same sign always + always opposite ( a − b = ( a − b ) a + ab + b 3 3 same sign always opposite 2 2 ) always +
6. 6. Now we know how to get the signs, let’s work on what goes inside. Square this term to get this term. ( a + b = ( a + b ) a − ab + b 3 3 2 2 ) Cube root of 1st term Cube root of 2nd term Product of cube root of 1st term and cube root of 2nd term.
7. 7. Try one. 27 x −125 = 3 Make a binomial and a trinomial with the correct signs.
8. 8. Try one. 27 x −125 = 3 ( − Cube root of 1st term )( + + ) Cube root of 2nd term
9. 9. Try one. 27 x −125 = ( 3x − 5)( 3 + + ) Square this term to get this term.
10. 10. Try one. 27 x −125 = ( 3x − 5) ( 9 x 2 + 3 + ) Multiply 3x an 5 to get this term.
11. 11. Try one. 27 x −125 = ( 3x − 5) ( 9 x + 15 x + 3 2 Square this term to get this term. )
12. 12. Try one. ( 27 x −125 = ( 3 x − 5) 9 x + 15 x + 25 3 2 ) You did it! Don’t forget the first rule of factoring is to look for the greatest common factor.