document.write( "Question 118297: How can you factor out, completely the problem 4t^2-12t+9-w^2? \n" ); document.write( "

Algebra.Com's Answer #86412 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"4t%5E2-12t%2B9-w%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( "Let's focus on the polynomial $\"4t%5E2-12t%2B9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"4t%5E2-12t%2B9\"$ we can see that the first term is $\"4t%5E2\"$ and the last term is $\"9\"$ where the coefficients are 4 and 9 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 4 and the last coefficient 9 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient -12? Let's list all of the factors of 36:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 36:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18 ...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 36\r
\n" ); document.write( "\n" ); document.write( "1*36\r
\n" ); document.write( "\n" ); document.write( "2*18\r
\n" ); document.write( "\n" ); document.write( "3*12\r
\n" ); document.write( "\n" ); document.write( "4*9\r
\n" ); document.write( "\n" ); document.write( "6*6\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-9)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-6)\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 -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12\r
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First Number	Second Number	Sum
1	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -6 and -6 add up to -12 and multiply to 36\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"4t%5E2-12t%2B9\"$ , replace $\"-12t\"$ with $\"-6t%2B-6t\"$ (notice $\"-6t%2B-6t\"$ adds up to $\"-12t\"$ . So it is equivalent to $\"-12t\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"4t%5E2%2Bhighlight%28-6t%2B-6t%29%2B9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"4t%5E2-6t-6t%2B9\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%284t%5E2-6t%29%2B%28-6t%2B9%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2t%282t-3%29-3%282t-3%29\"$ Factor out the GCF of $\"2t\"$ out of the first group. Factor out the GCF of $\"-3\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282t-3%29%282t-3%29\"$ Since we have a common term of $\"2t-3\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"4t%5E2-12t%2B9\"$ factors to $\"%282t-3%29%282t-3%29\"$ which is $\"%282t-3%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"4t%5E2-12t%2B9-w%5E2\"$ becomes $\"%282t-3%29%5E2-w%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Notice how we have a difference of squares. This means we can use the difference of squares formula to factor further.\r
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\n" ); document.write( "\n" ); document.write( "Let $\"A=2t-3\"$ and $\"B=w\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"A%5E2-B%5E2\"$ So we then get this\r
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\n" ); document.write( "\n" ); document.write( " $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ Now factor using the difference of squares\r
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\n" ); document.write( "\n" ); document.write( " $\"%282t-3%2Bw%29%282t-3-w%29\"$ Now replace A with $\"2t-3\"$ and B with $\"w\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4t%5E2-12t%2B9-w%5E2\"$ completely factors to $\"%282t-3%2Bw%29%282t-3-w%29\"$ \n" ); document.write( "

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