document.write( "Question 139589This question is from textbook Algebra Structure and Method
\n" ); document.write( ": I have been struggling with this problem for awhile now and I was wondering if anyone could help me? I would deeply appreciate it! Please and Thank you!\r
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\n" ); document.write( "\n" ); document.write( "Factoring Patterns for ax^2+bx+c
\n" ); document.write( "Factor. Check by multiplying the factors. If the polynomial is not factorable, write prime.\r
\n" ); document.write( "\n" ); document.write( "8+45r-18r^2 \n" ); document.write( "

Algebra.Com's Answer #101779 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"8%2B45r-18r%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"-18r%5E2%2B45r%2B8\"$ Sort the terms in descending order\r
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\n" ); document.write( "\n" ); document.write( " $\"-%2818r%5E2-45r-8%29\"$ Factor out a negative one\r
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\n" ); document.write( "\n" ); document.write( "Looking at the inner polynomial $\"18r%5E2-45r-8\"$ , we can see that the first term is $\"18r%5E2\"$ and the last term is -8\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 18 and the last term -8 to get -144. Now what two numbers multiply to -144 and add to the middle coefficient -45? Let's list all of the factors of -144:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -144:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,36,48,72\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...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 -144\r
\n" ); document.write( "\n" ); document.write( "(1)*(-144)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-72)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(8)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(144)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(72)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(48)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(36)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(18)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(16)\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 -45? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -45\r
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First Number	Second Number	Sum
1	-144	1+(-144)=-143
2	-72	2+(-72)=-70
3	-48	3+(-48)=-45
4	-36	4+(-36)=-32
6	-24	6+(-24)=-18
8	-18	8+(-18)=-10
9	-16	9+(-16)=-7
-1	144	-1+144=143
-2	72	-2+72=70
-3	48	-3+48=45
-4	36	-4+36=32
-6	24	-6+24=18
-8	18	-8+18=10
-9	16	-9+16=7

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 3 and -48 add up to -45 and multiply to -144\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"18r%5E2-45r-8\"$ , replace $\"-45r\"$ with $\"3r%2B-48r\"$ (notice $\"3r%2B-48r\"$ adds up to $\"-45r\"$ . So it is equivalent to $\"-45r\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"18r%5E2%2Bhighlight%283r%2B-48r%29%2B-8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"18r%5E2%2B3r-48r-8\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2818r%5E2%2B3r%29%2B%28-48r-8%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3r%286r%2B1%29-8%286r%2B1%29\"$ Factor out the GCF of $\"3r\"$ out of the first group. Factor out the GCF of $\"-8\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283r-8%29%286r%2B1%29\"$ Since we have a common term of $\"6r%2B1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"18r%5E2%2B3r-48r-8\"$ factors to $\"%283r-8%29%286r%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"18r%5E2-45r-8\"$ factors to $\"%283r-8%29%286r%2B1%29\"$ (since $\"18r%5E2-45r-8\"$ is equivalent to $\"18r%5E2%2B3r-48r-8\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So $\"-%2818r%5E2-45r-8%29\"$ factors to $\"-%283r-8%29%286r%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"8%2B45r-18r%5E2\"$ factors to $\"-%283r-8%29%286r%2B1%29\"$ \r
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