document.write( "Question 248456: I just learned this today and not sure how to do this. Please help. I have a quiz tomorrow on it. It is factoring. Thank you!!\r
\n" ); document.write( "\n" ); document.write( "56-15z+z^2 \n" ); document.write( "

Algebra.Com's Answer #181064 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"56-15z%2Bz%5E2\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"z%5E2-15z%2B56\"$ Rearrange the terms in descending degree.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"z%5E2-15z%2B56\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-15\"$ , and the last term is $\"56\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"56\"$ to get $\"%281%29%2856%29=56\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"56\"$ (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 $\"56\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"56\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,7,8,14,28,56\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-8,-14,-28,-56\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 $\"56\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*56 = 56
\n" ); document.write( "2*28 = 56
\n" ); document.write( "4*14 = 56
\n" ); document.write( "7*8 = 56
\n" ); document.write( "(-1)*(-56) = 56
\n" ); document.write( "(-2)*(-28) = 56
\n" ); document.write( "(-4)*(-14) = 56
\n" ); document.write( "(-7)*(-8) = 56\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	56	1+56=57
2	28	2+28=30
4	14	4+14=18
7	8	7+8=15
-1	-56	-1+(-56)=-57
-2	-28	-2+(-28)=-30
-4	-14	-4+(-14)=-18
-7	-8	-7+(-8)=-15

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-7\"$ and $\"-8\"$ add to $\"-15\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-7\"$ and $\"-8\"$ both multiply to $\"56\"$ and add to $\"-15\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-15z\"$ with $\"-7z-8z\"$ . Remember, $\"-7\"$ and $\"-8\"$ add to $\"-15\"$ . So this shows us that $\"-7z-8z=-15z\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"z%5E2%2Bhighlight%28-7z-8z%29%2B56\"$ Replace the second term $\"-15z\"$ with $\"-7z-8z\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28z%5E2-7z%29%2B%28-8z%2B56%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"z%28z-7%29%2B%28-8z%2B56%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"z%28z-7%29-8%28z-7%29\"$ Factor out $\"8\"$ 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( " $\"%28z-8%29%28z-7%29\"$ Combine like terms. Or factor out the common term $\"z-7\"$ \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 $\"56-15z%2Bz%5E2\"$ factors to $\"%28z-8%29%28z-7%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"56-15z%2Bz%5E2=%28z-8%29%28z-7%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28z-8%29%28z-7%29\"$ to get $\"z%5E2-15z%2B56\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );