document.write( "Question 156342: 8. Factoring is very hard for me. How do I complete this one?\r
\n" ); document.write( "\n" ); document.write( "15x^2 + 7x - 2\r
\n" ); document.write( "\n" ); document.write( "thank you for your help. \n" ); document.write( "

Algebra.Com's Answer #115129 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"15x%5E2%2B7x-2\"$ , we can see that the first coefficient is $\"15\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"15\"$ by the last term $\"-2\"$ to get $\"%2815%29%28-2%29=-30\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-30\"$ (the previous product) and add to the second coefficient $\"7\"$ ?\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 $\"-30\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-30\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,30\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30\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 $\"-30\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-30)
\n" ); document.write( "2*(-15)
\n" ); document.write( "3*(-10)
\n" ); document.write( "5*(-6)
\n" ); document.write( "(-1)*(30)
\n" ); document.write( "(-2)*(15)
\n" ); document.write( "(-3)*(10)
\n" ); document.write( "(-5)*(6)\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 $\"7\"$ :\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	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"10\"$ add to $\"7\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"10\"$ both multiply to $\"-30\"$ and add to $\"7\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"7x\"$ with $\"-3x%2B10x\"$ . Remember, $\"-3\"$ and $\"10\"$ add to $\"7\"$ . So this shows us that $\"-3x%2B10x=7x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"15x%5E2%2Bhighlight%28-3x%2B10x%29-2\"$ Replace the second term $\"7x\"$ with $\"-3x%2B10x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%2815x%5E2-3x%29%2B%2810x-2%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3x%285x-1%29%2B%2810x-2%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3x%285x-1%29%2B2%285x-1%29\"$ Factor out $\"2\"$ 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( " $\"%283x%2B2%29%285x-1%29\"$ Combine like terms. Or factor out the common term $\"5x-1\"$ \r
\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 $\"15x%5E2%2B7x-2\"$ factors to $\"%283x%2B2%29%285x-1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%283x%2B2%29%285x-1%29\"$ to get $\"15x%5E2%2B7x-2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );