document.write( "Question 200486: Hi all, just wondering if someone could show me how to factorise the following. Steps and explanations would be great.
\n" ); document.write( "z^2-5z-14\r
\n" ); document.write( "\n" ); document.write( "Thanks -Nick. \n" ); document.write( "

Algebra.Com's Answer #150762 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Looking at the expression $\"z%5E2-5z-14\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-14\"$ to get $\"%281%29%28-14%29=-14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-14\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-14\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-14\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,7,14\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-7,-14\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"-14\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-14)
\n" ); document.write( "2*(-7)
\n" ); document.write( "(-1)*(14)
\n" ); document.write( "(-2)*(7)\r
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\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 $\"-5\"$ :\r
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First Number	Second Number	Sum
1	-14	1+(-14)=-13
2	-7	2+(-7)=-5
-1	14	-1+14=13
-2	7	-2+7=5

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"-7\"$ add to $\"-5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"-7\"$ both multiply to $\"-14\"$ and add to $\"-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5z\"$ with $\"2z-7z\"$ . Remember, $\"2\"$ and $\"-7\"$ add to $\"-5\"$ . So this shows us that $\"2z-7z=-5z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"z%5E2%2Bhighlight%282z-7z%29-14\"$ Replace the second term $\"-5z\"$ with $\"2z-7z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28z%5E2%2B2z%29%2B%28-7z-14%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z%2B2%29%2B%28-7z-14%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z%2B2%29-7%28z%2B2%29\"$ Factor out $\"7\"$ 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
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\n" ); document.write( "\n" ); document.write( " $\"%28z-7%29%28z%2B2%29\"$ Combine like terms. Or factor out the common term $\"z%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"z%5E2-5z-14\"$ factors to $\"%28z-7%29%28z%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28z-7%29%28z%2B2%29\"$ to get $\"z%5E2-5z-14\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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