document.write( "Question 316714: how do i factor 4x^2+4x+1 \n" ); document.write( "
Algebra.Com's Answer #226652 by jim_thompson5910(35256)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "Looking at the expression \"4x%5E2%2B4x%2B1\", we can see that the first coefficient is \"4\", the second coefficient is \"4\", and the last term is \"1\".\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient \"4\" by the last term \"1\" to get \"%284%29%281%29=4\".\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to \"4\" (the previous product) and add to the second coefficient \"4\"?\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 \"4\" (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of \"4\":\r
\n" ); document.write( "\n" ); document.write( "1,2,4\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4\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 \"4\".\r
\n" ); document.write( "\n" ); document.write( "1*4 = 4
\n" ); document.write( "2*2 = 4
\n" ); document.write( "(-1)*(-4) = 4
\n" ); document.write( "(-2)*(-2) = 4\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 \"4\":\r
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First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4
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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers \"2\" and \"2\" add to \"4\" (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers \"2\" and \"2\" both multiply to \"4\" and add to \"4\"\r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term \"4x\" with \"2x%2B2x\". Remember, \"2\" and \"2\" add to \"4\". So this shows us that \"2x%2B2x=4x\".\r
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\n" ); document.write( "\n" ); document.write( "\"4x%5E2%2Bhighlight%282x%2B2x%29%2B1\" Replace the second term \"4x\" with \"2x%2B2x\".\r
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\n" ); document.write( "\n" ); document.write( "\"%284x%5E2%2B2x%29%2B%282x%2B1%29\" Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( "\"2x%282x%2B1%29%2B%282x%2B1%29\" Factor out the GCF \"2x\" from the first group.\r
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\n" ); document.write( "\n" ); document.write( "\"2x%282x%2B1%29%2B1%282x%2B1%29\" Factor out \"1\" 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( "\"%282x%2B1%29%282x%2B1%29\" Combine like terms. Or factor out the common term \"2x%2B1\"\r
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\n" ); document.write( "\n" ); document.write( "\"%282x%2B1%29%5E2\" Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So \"4x%5E2%2B4x%2B1\" factors to \"%282x%2B1%29%5E2\".\r
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\n" ); document.write( "\n" ); document.write( "In other words, \"4x%5E2%2B4x%2B1=%282x%2B1%29%5E2\".\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding \"%282x%2B1%29%5E2\" to get \"4x%5E2%2B4x%2B1\" or by graphing the original expression and the answer (the two graphs should be identical).
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