document.write( "Question 202196: Factor the following polynomial completely.\r
\n" ); document.write( "\n" ); document.write( "25a4 + 40a2 + 16 = \n" ); document.write( "

Algebra.Com's Answer #152455 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"25a%5E4%2B40a%5E2%2B16\"$ we can see that the first term is $\"25a%5E4\"$ and the last term is $\"16\"$ where the coefficients are 25 and 16 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 25 and the last coefficient 16 to get 400. Now what two numbers multiply to 400 and add to the middle coefficient 40? Let's list all of the factors of 400:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 400:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,16,20,25,40,50,80,100,200\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200 ...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 400\r
\n" ); document.write( "\n" ); document.write( "1*400\r
\n" ); document.write( "\n" ); document.write( "2*200\r
\n" ); document.write( "\n" ); document.write( "4*100\r
\n" ); document.write( "\n" ); document.write( "5*80\r
\n" ); document.write( "\n" ); document.write( "8*50\r
\n" ); document.write( "\n" ); document.write( "10*40\r
\n" ); document.write( "\n" ); document.write( "16*25\r
\n" ); document.write( "\n" ); document.write( "20*20\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-400)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-200)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-100)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-80)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(-50)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(-40)\r
\n" ); document.write( "\n" ); document.write( "(-16)*(-25)\r
\n" ); document.write( "\n" ); document.write( "(-20)*(-20)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 40? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 40\r
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First Number	Second Number	Sum
1	400	1+400=401
2	200	2+200=202
4	100	4+100=104
5	80	5+80=85
8	50	8+50=58
10	40	10+40=50
16	25	16+25=41
20	20	20+20=40
-1	-400	-1+(-400)=-401
-2	-200	-2+(-200)=-202
-4	-100	-4+(-100)=-104
-5	-80	-5+(-80)=-85
-8	-50	-8+(-50)=-58
-10	-40	-10+(-40)=-50
-16	-25	-16+(-25)=-41
-20	-20	-20+(-20)=-40

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 20 and 20 add up to 40 and multiply to 400\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"25a%5E4%2B40a%5E2%2B16\"$ , replace $\"40a%5E2\"$ with $\"20a%5E2%2B20a%5E2\"$ (notice $\"20a%5E2%2B20a%5E2\"$ adds up to $\"40a%5E2\"$ . So it is equivalent to $\"40a%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"25a%5E4%2Bhighlight%2820a%5E2%2B20a%5E2%29%2B16\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"25a%5E4%2B20a%5E2%2B20a%5E2%2B16\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2825a%5E4%2B20a%5E2%29%2B%2820a%5E2%2B16%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"5a%5E2%285a%5E2%2B4%29%2B4%285a%5E2%2B4%29\"$ Factor out the GCF of $\"5a%5E2\"$ out of the first group. Factor out the GCF of $\"4\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%285a%5E2%2B4%29%285a%5E2%2B4%29\"$ Since we have a common term of $\"5a%5E2%2B4\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"25a%5E4%2B20a%5E2%2B20a%5E2%2B16\"$ factors to $\"%285a%5E2%2B4%29%285a%5E2%2B4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"25a%5E4%2B40a%5E2%2B16\"$ factors to $\"%285a%5E2%2B4%29%285a%5E2%2B4%29\"$ (since $\"25a%5E4%2B40a%5E2%2B16\"$ is equivalent to $\"25a%5E4%2B20a%5E2%2B20a%5E2%2B16\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%285a%5E2%2B4%29%285a%5E2%2B4%29\"$ is equivalent to $\"%285a%5E2%2B4%29%5E2\"$ since the term $\"5a%5E2%2B4\"$ occurs twice. So $\"25a%5E4%2B40a%5E2%2B16\"$ also factors to $\"%285a%5E2%2B4%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"25a%5E4%2B40a%5E2%2B16\"$ factors to $\"%285a%5E2%2B4%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"25a%5E4%2B40a%5E2%2B16=%285a%5E2%2B4%29%5E2\"$ \r
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