document.write( "Question 581992: Factor Completely.
\n" ); document.write( "63b^2+294bf+343f^2 \n" ); document.write( "

Algebra.Com's Answer #371905 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"63b%5E2%2B294bf%2B343f%5E2\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"7%289b%5E2%2B42bf%2B49f%5E2%29\"$ Factor out the GCF $\"7\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"9b%5E2%2B42bf%2B49f%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9b%5E2%2B42bf%2B49f%5E2\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"42\"$ , and the last coefficient is $\"49\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last coefficient $\"49\"$ to get $\"%289%29%2849%29=441\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"441\"$ (the previous product) and add to the second coefficient $\"42\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"441\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"441\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,7,9,21,49,63,147,441\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-9,-21,-49,-63,-147,-441\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"441\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*441 = 441
\n" ); document.write( "3*147 = 441
\n" ); document.write( "7*63 = 441
\n" ); document.write( "9*49 = 441
\n" ); document.write( "21*21 = 441
\n" ); document.write( "(-1)*(-441) = 441
\n" ); document.write( "(-3)*(-147) = 441
\n" ); document.write( "(-7)*(-63) = 441
\n" ); document.write( "(-9)*(-49) = 441
\n" ); document.write( "(-21)*(-21) = 441\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"42\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	441	1+441=442
3	147	3+147=150
7	63	7+63=70
9	49	9+49=58
21	21	21+21=42
-1	-441	-1+(-441)=-442
-3	-147	-3+(-147)=-150
-7	-63	-7+(-63)=-70
-9	-49	-9+(-49)=-58
-21	-21	-21+(-21)=-42

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"21\"$ and $\"21\"$ add to $\"42\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"21\"$ and $\"21\"$ both multiply to $\"441\"$ and add to $\"42\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"42bf\"$ with $\"21bf%2B21bf\"$ . Remember, $\"21\"$ and $\"21\"$ add to $\"42\"$ . So this shows us that $\"21bf%2B21bf=42bf\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"9b%5E2%2Bhighlight%2821bf%2B21bf%29%2B49f%5E2\"$ Replace the second term $\"42bf\"$ with $\"21bf%2B21bf\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%289b%5E2%2B21bf%29%2B%2821bf%2B49f%5E2%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3b%283b%2B7f%29%2B%2821bf%2B49f%5E2%29\"$ Factor out the GCF $\"3b\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3b%283b%2B7f%29%2B7f%283b%2B7f%29\"$ Factor out $\"7f\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%283b%2B7f%29%283b%2B7f%29\"$ Combine like terms. Or factor out the common term $\"3b%2B7f\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%283b%2B7f%29%5E2\"$ Condense the terms.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"7%289b%5E2%2B42bf%2B49f%5E2%29\"$ then factors further to $\"7%283b%2B7f%29%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"63b%5E2%2B294bf%2B343f%5E2\"$ completely factors to $\"7%283b%2B7f%29%5E2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"63b%5E2%2B294bf%2B343f%5E2=7%283b%2B7f%29%5E2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"7%283b%2B7f%29%5E2\"$ to get $\"63b%5E2%2B294bf%2B343f%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );