document.write( "Question 387109: what is the factored form of 4k^3-13k^2-12k \n" ); document.write( "

Algebra.Com's Answer #273633 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"4k%5E3-13k%5E2-12k\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"k%284k%5E2-13k-12%29\"$ Factor out the GCF $\"k\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"4k%5E2-13k-12\"$ \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 $\"4k%5E2-13k-12\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"-12\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"-12\"$ to get $\"%284%29%28-12%29=-48\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-48\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?\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 $\"-48\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-48\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,48\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48\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 $\"-48\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-48) = -48
\n" ); document.write( "2*(-24) = -48
\n" ); document.write( "3*(-16) = -48
\n" ); document.write( "4*(-12) = -48
\n" ); document.write( "6*(-8) = -48
\n" ); document.write( "(-1)*(48) = -48
\n" ); document.write( "(-2)*(24) = -48
\n" ); document.write( "(-3)*(16) = -48
\n" ); document.write( "(-4)*(12) = -48
\n" ); document.write( "(-6)*(8) = -48\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 $\"-13\"$ :\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	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

\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 $\"-16\"$ add to $\"-13\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-16\"$ both multiply to $\"-48\"$ and add to $\"-13\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-13k\"$ with $\"3k-16k\"$ . Remember, $\"3\"$ and $\"-16\"$ add to $\"-13\"$ . So this shows us that $\"3k-16k=-13k\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4k%5E2%2Bhighlight%283k-16k%29-12\"$ Replace the second term $\"-13k\"$ with $\"3k-16k\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%284k%5E2%2B3k%29%2B%28-16k-12%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"k%284k%2B3%29%2B%28-16k-12%29\"$ Factor out the GCF $\"k\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"k%284k%2B3%29-4%284k%2B3%29\"$ 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( " $\"%28k-4%29%284k%2B3%29\"$ Combine like terms. Or factor out the common term $\"4k%2B3\"$ \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 $\"k%284k%5E2-13k-12%29\"$ then factors further to $\"k%28k-4%29%284k%2B3%29\"$ \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 $\"4k%5E3-13k%5E2-12k\"$ completely factors to $\"k%28k-4%29%284k%2B3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"4k%5E3-13k%5E2-12k=k%28k-4%29%284k%2B3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"k%28k-4%29%284k%2B3%29\"$ to get $\"4k%5E3-13k%5E2-12k\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, feel free to check out my tutoring website\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

\n" );