document.write( "Question 164436: Please help Factor Completely. If the polynomial is prime, please state this.
\n" ); document.write( "a^2-13a+40 \n" ); document.write( "

Algebra.Com's Answer #121132 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"a%5E2-13a%2B40\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"40\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"40\"$ to get $\"%281%29%2840%29=40\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"40\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?\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 $\"40\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"40\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,20,40\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-20,-40\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 $\"40\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*40
\n" ); document.write( "2*20
\n" ); document.write( "4*10
\n" ); document.write( "5*8
\n" ); document.write( "(-1)*(-40)
\n" ); document.write( "(-2)*(-20)
\n" ); document.write( "(-4)*(-10)
\n" ); document.write( "(-5)*(-8)\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 $\"-13\"$ :\r
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First Number	Second Number	Sum
1	40	1+40=41
2	20	2+20=22
4	10	4+10=14
5	8	5+8=13
-1	-40	-1+(-40)=-41
-2	-20	-2+(-20)=-22
-4	-10	-4+(-10)=-14
-5	-8	-5+(-8)=-13

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-5\"$ and $\"-8\"$ add to $\"-13\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-5\"$ and $\"-8\"$ both multiply to $\"40\"$ and add to $\"-13\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-13a\"$ with $\"-5a-8a\"$ . Remember, $\"-5\"$ and $\"-8\"$ add to $\"-13\"$ . So this shows us that $\"-5a-8a=-13a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E2%2Bhighlight%28-5a-8a%29%2B40\"$ Replace the second term $\"-13a\"$ with $\"-5a-8a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2-5a%29%2B%28-8a%2B40%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a-5%29%2B%28-8a%2B40%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a-5%29-8%28a-5%29\"$ 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
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\n" ); document.write( "\n" ); document.write( " $\"%28a-8%29%28a-5%29\"$ Combine like terms. Or factor out the common term $\"a-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"a%5E2-13a%2B40\"$ factors to $\"%28a-8%29%28a-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28a-8%29%28a-5%29\"$ to get $\"a%5E2-13a%2B40\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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