document.write( "Question 168695: Factor....Need some help\r
\n" ); document.write( "\n" ); document.write( "x^4-13x^2+36 \n" ); document.write( "

Algebra.Com's Answer #124346 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E4-13x%5E2%2B36\"$ we can see that the first term is $\"1x%5E4\"$ and the last term is $\"36\"$ where the coefficients are 1 and 36 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 36 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient -13? Let's list all of the factors of 36:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of 36:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18\r
\n" ); document.write( "
\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
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -13\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

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

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that -4 and -9 add up to -13 and multiply to 36\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"1x%5E4-13x%5E2%2B36\"$ , replace $\"-13x%5E2\"$ with $\"-4x%5E2%2B-9x%5E2\"$ (notice $\"-4x%5E2%2B-9x%5E2\"$ adds up to $\"-13x%5E2\"$ . So it is equivalent to $\"-13x%5E2\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E4%2Bhighlight%28-4x%5E2%2B-9x%5E2%29%2B36\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E4-4x%5E2-9x%5E2%2B36\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E4-4x%5E2%29%2B%28-9x%5E2%2B36%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%28x%5E2-4%29-9%28x%5E2-4%29\"$ Factor out the GCF of $\"x%5E2\"$ out of the first group. Factor out the GCF of $\"-9\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-9%29%28x%5E2-4%29\"$ Since we have a common term of $\"x%5E2-4\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E4-4x%5E2-9x%5E2%2B36\"$ factors to $\"%28x%5E2-9%29%28x%5E2-4%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E4-13x%5E2%2B36\"$ factors to $\"%28x%5E2-9%29%28x%5E2-4%29\"$ (since $\"x%5E4-13x%5E2%2B36\"$ is equivalent to $\"x%5E4-4x%5E2-9x%5E2%2B36\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E4-13x%5E2%2B36\"$ factors to $\"%28x%5E2-9%29%28x%5E2-4%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-----------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-9%29%28x%5E2-4%29\"$ Start with the given factorization\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B3%29%28x-3%29%28x%5E2-4%29\"$ Factor $\"x%5E2-9\"$ to get $\"%28x%2B3%29%28x-3%29\"$ (by use of the difference of squares)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B3%29%28x-3%29%28x%2B2%29%28x-2%29\"$ Factor $\"x%5E2-4\"$ to get $\"%28x%2B2%29%28x-2%29\"$ (by use of the difference of squares)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E4-13x%5E2%2B36\"$ completely factors to $\"%28x%2B3%29%28x-3%29%28x%2B2%29%28x-2%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: the order of the factors does not matter. \n" ); document.write( "

\n" );