document.write( "Question 142171: Factor the following 3z^2-10z-25 \n" ); document.write( "

Algebra.Com's Answer #103519 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"3z%5E2-10z-25\"$ we can see that the first term is $\"3z%5E2\"$ and the last term is $\"-25\"$ where the coefficients are 3 and -25 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient -25 to get -75. Now what two numbers multiply to -75 and add to the middle coefficient -10? Let's list all of the factors of -75:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -75:\r
\n" ); document.write( "\n" ); document.write( "1,3,5,15,25,75\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-15,-25,-75 ...List the negative factors as well. This will allow us to find all possible combinations\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -75\r
\n" ); document.write( "\n" ); document.write( "(1)*(-75)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-25)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(75)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(25)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(15)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -10\r
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First Number	Second Number	Sum
1	-75	1+(-75)=-74
3	-25	3+(-25)=-22
5	-15	5+(-15)=-10
-1	75	-1+75=74
-3	25	-3+25=22
-5	15	-5+15=10

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 5 and -15 add up to -10 and multiply to -75\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"3z%5E2-10z-25\"$ , replace $\"-10z\"$ with $\"5z%2B-15z\"$ (notice $\"5z%2B-15z\"$ adds up to $\"-10z\"$ . So it is equivalent to $\"-10z\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"3z%5E2%2Bhighlight%285z%2B-15z%29%2B-25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"3z%5E2%2B5z-15z-25\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%283z%5E2%2B5z%29%2B%28-15z-25%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"z%283z%2B5%29-5%283z%2B5%29\"$ Factor out the GCF of $\"z\"$ out of the first group. Factor out the GCF of $\"-5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28z-5%29%283z%2B5%29\"$ Since we have a common term of $\"3z%2B5\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"3z%5E2%2B5z-15z-25\"$ factors to $\"%28z-5%29%283z%2B5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"3z%5E2-10z-25\"$ factors to $\"%28z-5%29%283z%2B5%29\"$ (since $\"3z%5E2-10z-25\"$ is equivalent to $\"3z%5E2%2B5z-15z-25\"$ )\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3z%5E2-10z-25\"$ factors to $\"%28z-5%29%283z%2B5%29\"$ \r
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