document.write( "Question 317784: factorize 6x^2+17x+5 by splitting middle term snd by using the factor theorm \n" ); document.write( "

Algebra.Com's Answer #227548 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2%2B17x%2B5\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"17\"$ , and the last term is $\"5\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"5\"$ to get $\"%286%29%285%29=30\"$ .\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 $\"17\"$ ?\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 = 30
\n" ); document.write( "2*15 = 30
\n" ); document.write( "3*10 = 30
\n" ); document.write( "5*6 = 30
\n" ); document.write( "(-1)*(-30) = 30
\n" ); document.write( "(-2)*(-15) = 30
\n" ); document.write( "(-3)*(-10) = 30
\n" ); document.write( "(-5)*(-6) = 30\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 $\"17\"$ :\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=31
2	15	2+15=17
3	10	3+10=13
5	6	5+6=11
-1	-30	-1+(-30)=-31
-2	-15	-2+(-15)=-17
-3	-10	-3+(-10)=-13
-5	-6	-5+(-6)=-11

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"15\"$ add to $\"17\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"15\"$ both multiply to $\"30\"$ and add to $\"17\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"17x\"$ with $\"2x%2B15x\"$ . Remember, $\"2\"$ and $\"15\"$ add to $\"17\"$ . So this shows us that $\"2x%2B15x=17x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%282x%2B15x%29%2B5\"$ Replace the second term $\"17x\"$ with $\"2x%2B15x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2%2B2x%29%2B%2815x%2B5%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%283x%2B1%29%2B%2815x%2B5%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%283x%2B1%29%2B5%283x%2B1%29\"$ Factor out $\"5\"$ 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( " $\"%282x%2B5%29%283x%2B1%29\"$ Combine like terms. Or factor out the common term $\"3x%2B1\"$ \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 $\"6x%5E2%2B17x%2B5\"$ factors to $\"%282x%2B5%29%283x%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E2%2B17x%2B5=%282x%2B5%29%283x%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282x%2B5%29%283x%2B1%29\"$ to get $\"6x%5E2%2B17x%2B5\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );