Sum or difference of Cubes
Factoring Cubes 5 practice problems EVERY time you look at a factoring problem, start with the question: Is there a GCF? What s a GCF? It s a number or a variable that all the terms in the problem have in common, that you divide out of all of them before you get started. We ll use it a LOT in section 5., some in 5.4, not much in this part of 5.5. Nevertheless, we ll keep asking ourselves that question. :-D
Cubes: 1 8 7 64 15 a 15 Start this game the same way we ve been playing: Is there a GCF? No, so count the terms. There are, so this is one of those weird ones!!! Since the exponent is a, this must be a cube problem: a SOAP. Now, all changes. One regular set of parenthesis, one larger. Also, write down the cube root of each of those two terms, then cross out the original terms. a 15 a 5 ( )( ) The terms a and 5 are your little building blocks. You ll build your answer from those. First, put them in the first set of parenthesis, along with the original sign in your problem. ( a 5)( ) Fill out the second set of parenthesis on the next page.
SOAP Same Opposite Always Positive The second set of parenthesis has terms in it. ( a 5)( ) In the first spot goes the first term times itself. ( a 5)( a ) In the back spot goes the last term times itself. ( a 5)( a 5) And in that middle spot goes the two terms times each other. ( a 5)( a 5a 5) Now put the correct signs in. SOAP this problem! ( a 5)( a 5a 5) ( a 5)( a 5a 5)
Cubes: 1 8 7 64 15 x 8 Start this game the same way we ve been playing: Is there a GCF? No, so count the terms. There are, so this is one of those weird ones!!! Since the exponent is a, this must be a cube problem: a SOAP. Now, all changes. One regular set of parenthesis, one larger. Also, write down the cube root of each of those two terms, then cross out the original terms. x 8 x ( )( ) The terms x and are your little building blocks. You ll build your answer from those. First, put them in the first set of parenthesis, along with the original sign in your problem. ( x )( ) Fill out the second set of parenthesis on the next page.
SOAP Same Opposite Always Positive The second set of parenthesis has terms in it. ( x )( ) In the first spot goes the first term times itself. ( x )( x ) In the back spot goes the last term times itself. ( x )( x 4) And in that middle spot goes the two terms times each other. ( x )( x x 4) Now put the correct signs in. SOAP this problem! ( x )( x x 4) ( x )( x x 4)
Cubes: 1 8 7 64 15 7x 1 Start this game the same way we ve been playing: Is there a GCF? No, so count the terms. There are, so this is one of those weird ones!!! Since the exponent is a, this must be a cube problem: a SOAP. Now, all changes. One regular set of parenthesis, one larger. Also, write down the cube root of each of those two terms, then cross out the original terms. 7x 1 x 1 ( )( ) The terms x and 1 are your little building blocks. You ll build your answer from those. First, put them in the first set of parenthesis, along with the original sign in your problem. (x 1)( ) Fill out the second set of parenthesis on the next page.
SOAP Same Opposite Always Positive The second set of parenthesis has terms in it. (x 1)( ) In the first spot goes the first term times itself. (x 1)(9 x ) In the back spot goes the last term times itself. (x 1)(9 x 1) And in that middle spot goes the two terms times each other. (x 1)(9 x x 1) Now put the correct signs in. SOAP this problem! (x 1)(9 x x 1) (x 1)(9 x x 1)
Cubes: 1 8 7 64 15 a 64 Start this game the same way we ve been playing: Is there a GCF? No, so count the terms. There are, so this is one of those weird ones!!! Since the exponent is a, this must be a cube problem: a SOAP. Now, all changes. One regular set of parenthesis, one larger. Also, write down the cube root of each of those two terms, then cross out the original terms. a 64 a 4 ( )( ) The terms a and 4 are your little building blocks. You ll build your answer from those. First, put them in the first set of parenthesis, along with the original sign in your problem. ( a 4)( ) Fill out the second set of parenthesis on the next page.
SOAP Same Opposite Always Positive The second set of parenthesis has terms in it. ( a 4)( ) In the first spot goes the first term times itself. ( a 4)( a ) In the back spot goes the last term times itself. ( a 4)( a 16) And in that middle spot goes the two terms times each other. ( a 4)( a 4a 16) Now put the correct signs in. SOAP this problem! ( a 4)( a 4a 16) ( a 4)( a 4a 16)
Cubes: 1 8 7 64 15 7y 8 Start this game the same way we ve been playing: Is there a GCF? No, so count the terms. There are, so this is one of those weird ones!!! Since the exponent is a, this must be a cube problem: a SOAP. Now, all changes. One regular set of parenthesis, one larger. Also, write down the cube root of each of those two terms, then cross out the original terms. 7y 8 y ( )( ) The terms y and are your little building blocks. You ll build your answer from those. First, put them in the first set of parenthesis, along with the original sign in your problem. (y )( ) Fill out the second set of parenthesis on the next page.
SOAP Same Opposite Always Positive The second set of parenthesis has terms in it. (y )( ) In the first spot goes the first term times itself. (y )(9 y ) In the back spot goes the last term times itself. (y )(9 y 4) And in that middle spot goes the two terms times each other. (y )(9 y 6y 4) Now put the correct signs in. SOAP this problem! (y )(9 y 6y 4) (y )(9 y 6y 4)