document.write( "Question 243710: Here is the problem I don't understand.\r
\n" ); document.write( "\n" ); document.write( "Factor this trinomial
\n" ); document.write( "-45p2 - 12p + 12p3 \n" ); document.write( "

Algebra.Com's Answer #178511 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"-45p%5E2+-+12p+%2B+12p%5E3\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"12p%5E3-45p%5E2+-+12p+\"$ Rearrange the terms in descending degree.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3p%284p%5E2-15p-4%29\"$ Factor out the GCF $\"3p\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"4p%5E2-15p-4\"$ \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 $\"4p%5E2-15p-4\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-15\"$ , and the last term is $\"-4\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"-4\"$ to get $\"%284%29%28-4%29=-16\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-16\"$ (the previous product) and add to the second coefficient $\"-15\"$ ?\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 $\"-16\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-16\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8,16\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8,-16\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 $\"-16\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-16) = -16
\n" ); document.write( "2*(-8) = -16
\n" ); document.write( "4*(-4) = -16
\n" ); document.write( "(-1)*(16) = -16
\n" ); document.write( "(-2)*(8) = -16
\n" ); document.write( "(-4)*(4) = -16\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 $\"-15\"$ :\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	-16	1+(-16)=-15
2	-8	2+(-8)=-6
4	-4	4+(-4)=0
-1	16	-1+16=15
-2	8	-2+8=6
-4	4	-4+4=0

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"1\"$ and $\"-16\"$ add to $\"-15\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"1\"$ and $\"-16\"$ both multiply to $\"-16\"$ and add to $\"-15\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-15p\"$ with $\"p-16p\"$ . Remember, $\"1\"$ and $\"-16\"$ add to $\"-15\"$ . So this shows us that $\"p-16p=-15p\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4p%5E2%2Bhighlight%28p-16p%29-4\"$ Replace the second term $\"-15p\"$ with $\"p-16p\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%284p%5E2%2Bp%29%2B%28-16p-4%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"p%284p%2B1%29%2B%28-16p-4%29\"$ Factor out the GCF $\"p\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"p%284p%2B1%29-4%284p%2B1%29\"$ Factor out $\"4\"$ 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( " $\"%28p-4%29%284p%2B1%29\"$ Combine like terms. Or factor out the common term $\"4p%2B1\"$ \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( "So $\"3p%284p%5E2-15p-4%29\"$ then factors further to $\"3p%28p-4%29%284p%2B1%29\"$ \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 $\"-45p%5E2+-+12p+%2B+12p%5E3\"$ completely factors to $\"3p%28p-4%29%284p%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"-45p%5E2+-+12p+%2B+12p%5E3=3p%28p-4%29%284p%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3p%28p-4%29%284p%2B1%29\"$ to get $\"-45p%5E2+-+12p+%2B+12p%5E3\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );