document.write( "Question 423054: How do I simplify the trinomial 3x^2+10x+3 into a binomial? \n" ); document.write( "

Algebra.Com's Answer #295177 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you mean factor into a product of binomials.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3x%5E2%2B10x%2B3\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"10\"$ , and the last term is $\"3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"3\"$ to get $\"%283%29%283%29=9\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"9\"$ (the previous product) and add to the second coefficient $\"10\"$ ?\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 $\"9\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"9\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,9\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-9\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 $\"9\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*9 = 9
\n" ); document.write( "3*3 = 9
\n" ); document.write( "(-1)*(-9) = 9
\n" ); document.write( "(-3)*(-3) = 9\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 $\"10\"$ :\r
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First Number	Second Number	Sum
1	9	1+9=10
3	3	3+3=6
-1	-9	-1+(-9)=-10
-3	-3	-3+(-3)=-6

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"1\"$ and $\"9\"$ add to $\"10\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"1\"$ and $\"9\"$ both multiply to $\"9\"$ and add to $\"10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"10x\"$ with $\"x%2B9x\"$ . Remember, $\"1\"$ and $\"9\"$ add to $\"10\"$ . So this shows us that $\"x%2B9x=10x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28x%2B9x%29%2B3\"$ Replace the second term $\"10x\"$ with $\"x%2B9x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2%2Bx%29%2B%289x%2B3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x%2B1%29%2B%289x%2B3%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x%2B1%29%2B3%283x%2B1%29\"$ Factor out $\"3\"$ 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( " $\"%28x%2B3%29%283x%2B1%29\"$ Combine like terms. Or factor out the common term $\"3x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B10x%2B3\"$ factors to $\"%28x%2B3%29%283x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2%2B10x%2B3=%28x%2B3%29%283x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B3%29%283x%2B1%29\"$ to get $\"3x%5E2%2B10x%2B3\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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