document.write( "Question 625095: Factorise : (a-2b)^3-512b^3 \n" ); document.write( "
Algebra.Com's Answer #393305 by jsmallt9(3759)\"\" \"About 
You can put this solution on YOUR website!
\"%28a-2b%29%5E3-512b%5E3\"
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  • Among the factoring techniques you have learned is factoring by patterns.
  • Among the patterns you should know are:
    1. \"p%5E2-q%5E2+=+%28p%2Bq%29%28p-q%29\"
    2. \"p%5E3%2Bq%5E3+=+%28p%2Bq%29%28p%5E2-pq%2Bq%5E2%29\"
    3. \"p%5E3-q%5E3+=+%28p-q%29%28p%5E2%2Bpq%2Bq%5E2%29\"
    4. \"p%5E2%2B2pq%2Bq%5E2+=+%28p%2Bq%29%28p%2Bq%29+=+%28p%2Bq%29%5E2\"
    5. \"p%5E2-2pq%2Bq%5E2+=+%28p-q%29%28p-q%29+=+%28p-q%29%5E2\"
  • The most difficult thing to master when learning to factor with these patterns is learning that the p's and q's in these formulas can be replaced with any valid mathematical expression and the pattern will still be true!! For example, you may have learned these patterns with different letters in them. Maybe a's and b's instead of p'q and q's. Or maybe x's and y's instead of the p's and q's. The point is that these are patterns. The specifics of what letters (or expressions) that takes the place of the p's and q's don't matter! For example, using the first pattern:
    \n" ); document.write( "\"p%5E2-q%5E2+=+%28p%2Bq%29%28p-q%29\"
    \n" ); document.write( "we could write:
    \n" ); document.write( "\"a%5E2-b%5E2+=+%28a%2Bb%29%28a-b%29\"
    \n" ); document.write( "Note how each \"p\" has been replaced with an \"a\" and each \"q\" with a \"b\". Or:
    \n" ); document.write( "\"x%5E2-1%5E2+=+%28x%2B1%29%28x-1%29\"
    \n" ); document.write( "Here's the p's have been replaced by x's and the q's with 1's. Or
    \n" ); document.write( "\"%28x%5E2%29%5E2-%28y%5E3%29%5E2+=+%28x%5E2%2By%5E3%29%28x%5E2-y%5E3%29\"
    \n" ); document.write( "Here the p's have been replaced with \"x%5E2\"'s and the q's with \"y%5E3\"'s. One more:
    \n" ); document.write( "\"%28j-k%29%5E2-%28c%2B3d%29%5E2+=+%28%28j-k%29%2B%28c%2B3d%29%29%28%28j-k%29-%28c%2B3d%29%29\"
    \n" ); document.write( "with the p's replaced by (j-k)'s and the q's replaced by (c+3d)'s. Once you get how these patterns work, it becomes much easier to factor with them.
Our expression nearly fits the pattern of pattern #3 above. It has two terms, with a minus between them and the first term is clearly a perfect cube. The difference between the pattern and our expression is that the second term of our expression is not a perfect cube... at least not yet.

\n" ); document.write( "Is \"512b%5E3\" the result of cubing something? Clearly the \"b%5E3\" is a cube of b. But what about the 512? Is it the cube of something? The answer, which we can find with a little trial and error, is yes. \"512+=+8%5E3\". So we can rewrite our expression as:
\n" ); document.write( "\"%28a-2b%29%5E3-%288b%29%5E3\"
\n" ); document.write( "Our expression now matches the pattern exactly, with the p's replaced with (a-2b)'s and the q's replaced by 8b. So we can use the pattern to factor it. We just write the right side pattern replacing the p's with (a-2b)'s and the q's with 8b's:
\n" ); document.write( "\"%28%28a-2b%29-8b%29%28%28a-2b%29%5E2%2B%28a-2b%29%288b%29+%2B+%288b%29%5E2%29\"
\n" ); document.write( "All we have left is simplifying. To square \"%28a-2b%29%5E2\" we can use pattern #5! This matches the right side of the pattern with the \"p\" replaced by \"a\" and the \"q\" with \"2b\". We can square this quickly by writing the left side of that pattern with the p's and q's repalced by a's and 2b's. The other parts of the simplifying are relative easy:
\n" ); document.write( "\"%28a-10b%29%28a%5E2-2%28a%29%282b%29%2B%282b%29%5E2+%2B+8ab+-+16b%5E2%2B64b%5E2%29\"
\n" ); document.write( "\"%28a-10b%29%28a%5E2-4ab%2B4b%5E2+%2B+8ab+-+16b%5E2+%2B+64b%5E2%29\"
\n" ); document.write( "\"%28a-10b%29%28a%5E2%2B4ab%2B52b%5E2%29\"
\n" ); document.write( "Since this will not factor any further, this is the fully-factored form of \"%28a-2b%29%5E3-512b%5E3\".

\n" ); document.write( "The main alternative to factoring with this pattern would be to cube (a-2b) (which is not easy):
\n" ); document.write( "\"%28a-2b%29%5E3-512b%5E3\"
\n" ); document.write( "\"a%5E3-6a%5E2%2Ab%2B12ab%5E2-8b%5E3-512b%5E3\"
\n" ); document.write( "Add like terms
\n" ); document.write( "\"a%5E3-6a%5E2%2Ab%2B12ab%5E2-520b%5E3\"
\n" ); document.write( "and then try to factor this. Factoring this is much more difficult than using the pattern. You probably have to find a clever way to factor by grouping or a clever way to factor using possible rational roots.

\n" ); document.write( "P.S. You may notice that when I replaced the p's and q's in the patterns, sometimes I used parentheses and sometimes I didn't. If you are confused about when to use parentheses and when not to, then just use parentheses all the time!. It can't hurt to use them when they are not needed. And it will really mess things up if you don't use them when you should. So if there is any doubt, then use parentheses! \n" ); document.write( "
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