document.write( "Question 133168: How do you factor the following problems step by step?\r
\n" ); document.write( "\n" ); document.write( "1. 9-x^2
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\n" ); document.write( "3. 4y^2+16y+16
\n" ); document.write( "4. x^-25/x+5 \n" ); document.write( "

Algebra.Com's Answer #97376 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll do the first two to get you started\r
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\n" ); document.write( "\n" ); document.write( " $\"9-x%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"%283%29%5E2-x%5E2\"$ Rewrite $\"9\"$ as $\"%283%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%283%29%5E2-%28x%29%5E2\"$ Rewrite $\"x%5E2\"$ as $\"%28x%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now use the difference of squares. Remember, the difference of squares formula is $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ where in this case $\"A=3\"$ and $\"B=x\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"9-x%5E2=%283%2Bx%29%283-x%29\"$ Plug in $\"A=3\"$ and $\"B=x\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So the expression\r
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\n" ); document.write( "\n" ); document.write( " $\"9-x%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "factors to\r
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\n" ); document.write( "\n" ); document.write( " $\"%283%2Bx%29%283-x%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Notice that if you foil the factored expression, you get the original expression. This verifies our answer.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2B81-36x\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2-36x%2B81\"$ Rearrange the terms\r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"4x%5E2-36x%2B81\"$ we can see that the first term is $\"4x%5E2\"$ and the last term is $\"81\"$ where the coefficients are 4 and 81 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 4 and the last coefficient 81 to get 324. Now what two numbers multiply to 324 and add to the middle coefficient -36? Let's list all of the factors of 324:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 324:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,27,36,54,81,108,162\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162 ...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 324\r
\n" ); document.write( "\n" ); document.write( "1*324\r
\n" ); document.write( "\n" ); document.write( "2*162\r
\n" ); document.write( "\n" ); document.write( "3*108\r
\n" ); document.write( "\n" ); document.write( "4*81\r
\n" ); document.write( "\n" ); document.write( "6*54\r
\n" ); document.write( "\n" ); document.write( "9*36\r
\n" ); document.write( "\n" ); document.write( "12*27\r
\n" ); document.write( "\n" ); document.write( "18*18\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-324)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-162)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-108)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-81)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-54)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(-27)\r
\n" ); document.write( "\n" ); document.write( "(-18)*(-18)\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 -36? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -36\r
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First Number	Second Number	Sum
1	324	1+324=325
2	162	2+162=164
3	108	3+108=111
4	81	4+81=85
6	54	6+54=60
9	36	9+36=45
12	27	12+27=39
18	18	18+18=36
-1	-324	-1+(-324)=-325
-2	-162	-2+(-162)=-164
-3	-108	-3+(-108)=-111
-4	-81	-4+(-81)=-85
-6	-54	-6+(-54)=-60
-9	-36	-9+(-36)=-45
-12	-27	-12+(-27)=-39
-18	-18	-18+(-18)=-36

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -18 and -18 add up to -36 and multiply to 324\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"4x%5E2-36x%2B81\"$ , replace $\"-36x\"$ with $\"-18x%2B-18x\"$ (notice $\"-18x%2B-18x\"$ adds up to $\"-36x\"$ . So it is equivalent to $\"-36x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%28-18x%2B-18x%29%2B81\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"4x%5E2-18x-18x%2B81\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2-18x%29%2B%28-18x%2B81%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%282x-9%29-9%282x-9%29\"$ Factor out the GCF of $\"2x\"$ out of the first group. Factor out the GCF of $\"-9\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x-9%29%282x-9%29\"$ Since we have a common term of $\"2x-9\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-18x-18x%2B81\"$ factors to $\"%282x-9%29%282x-9%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"4x%5E2-36x%2B81\"$ factors to $\"%282x-9%29%282x-9%29\"$ (since $\"4x%5E2-36x%2B81\"$ is equivalent to $\"4x%5E2-18x-18x%2B81\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%282x-9%29%282x-9%29\"$ is equivalent to $\"%282x-9%29%5E2\"$ since the term $\"2x-9\"$ occurs twice. So $\"4x%5E2-36x%2B81\"$ also factors to $\"%282x-9%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-36x%2B81\"$ factors to $\"%282x-9%29%5E2\"$ \r
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