document.write( "Question 416132: Factor the expression correctly 32b^2 - 20b - 3. i dont know how to factor it if all the numbers have no GCF. Can u please help me how to factor it. \n" ); document.write( "

Algebra.Com's Answer #291604 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"32b%5E2-20b-3\"$ , we can see that the first coefficient is $\"32\"$ , the second coefficient is $\"-20\"$ , and the last term is $\"-3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"32\"$ by the last term $\"-3\"$ to get $\"%2832%29%28-3%29=-96\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-96\"$ (the previous product) and add to the second coefficient $\"-20\"$ ?\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 $\"-96\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-96\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,32,48,96\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96\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 $\"-96\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-96) = -96
\n" ); document.write( "2*(-48) = -96
\n" ); document.write( "3*(-32) = -96
\n" ); document.write( "4*(-24) = -96
\n" ); document.write( "6*(-16) = -96
\n" ); document.write( "8*(-12) = -96
\n" ); document.write( "(-1)*(96) = -96
\n" ); document.write( "(-2)*(48) = -96
\n" ); document.write( "(-3)*(32) = -96
\n" ); document.write( "(-4)*(24) = -96
\n" ); document.write( "(-6)*(16) = -96
\n" ); document.write( "(-8)*(12) = -96\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 $\"-20\"$ :\r
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First Number	Second Number	Sum
1	-96	1+(-96)=-95
2	-48	2+(-48)=-46
3	-32	3+(-32)=-29
4	-24	4+(-24)=-20
6	-16	6+(-16)=-10
8	-12	8+(-12)=-4
-1	96	-1+96=95
-2	48	-2+48=46
-3	32	-3+32=29
-4	24	-4+24=20
-6	16	-6+16=10
-8	12	-8+12=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"4\"$ and $\"-24\"$ add to $\"-20\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"4\"$ and $\"-24\"$ both multiply to $\"-96\"$ and add to $\"-20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-20b\"$ with $\"4b-24b\"$ . Remember, $\"4\"$ and $\"-24\"$ add to $\"-20\"$ . So this shows us that $\"4b-24b=-20b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"32b%5E2%2Bhighlight%284b-24b%29-3\"$ Replace the second term $\"-20b\"$ with $\"4b-24b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2832b%5E2%2B4b%29%2B%28-24b-3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"4b%288b%2B1%29%2B%28-24b-3%29\"$ Factor out the GCF $\"4b\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"4b%288b%2B1%29-3%288b%2B1%29\"$ 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( " $\"%284b-3%29%288b%2B1%29\"$ Combine like terms. Or factor out the common term $\"8b%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"32b%5E2-20b-3\"$ factors to $\"%284b-3%29%288b%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"32b%5E2-20b-3=%284b-3%29%288b%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%284b-3%29%288b%2B1%29\"$ to get $\"32b%5E2-20b-3\"$ 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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