Question 116761
Do you want to factor this?




Looking at {{{n^2-36n+320}}} we can see that the first term is {{{n^2}}} and the last term is {{{320}}} where the coefficients are 1 and 320 respectively.


Now multiply the first coefficient 1 and the last coefficient 320 to get 320. Now what two numbers multiply to 320 and add to the  middle coefficient -36? Let's list all of the factors of 320:




Factors of 320:

1,2,4,5,8,10,16,20,32,40,64,80,160,320


-1,-2,-4,-5,-8,-10,-16,-20,-32,-40,-64,-80,-160,-320 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 320

1*320

2*160

4*80

5*64

8*40

10*32

16*20

(-1)*(-320)

(-2)*(-160)

(-4)*(-80)

(-5)*(-64)

(-8)*(-40)

(-10)*(-32)

(-16)*(-20)


note: remember two negative numbers multiplied together make a positive number



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


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">320</td><td>1+320=321</td></tr><tr><td align="center">2</td><td align="center">160</td><td>2+160=162</td></tr><tr><td align="center">4</td><td align="center">80</td><td>4+80=84</td></tr><tr><td align="center">5</td><td align="center">64</td><td>5+64=69</td></tr><tr><td align="center">8</td><td align="center">40</td><td>8+40=48</td></tr><tr><td align="center">10</td><td align="center">32</td><td>10+32=42</td></tr><tr><td align="center">16</td><td align="center">20</td><td>16+20=36</td></tr><tr><td align="center">-1</td><td align="center">-320</td><td>-1+(-320)=-321</td></tr><tr><td align="center">-2</td><td align="center">-160</td><td>-2+(-160)=-162</td></tr><tr><td align="center">-4</td><td align="center">-80</td><td>-4+(-80)=-84</td></tr><tr><td align="center">-5</td><td align="center">-64</td><td>-5+(-64)=-69</td></tr><tr><td align="center">-8</td><td align="center">-40</td><td>-8+(-40)=-48</td></tr><tr><td align="center">-10</td><td align="center">-32</td><td>-10+(-32)=-42</td></tr><tr><td align="center">-16</td><td align="center">-20</td><td>-16+(-20)=-36</td></tr></table>



From this list we can see that -16 and -20 add up to -36 and multiply to 320



Now looking at the expression {{{n^2-36n+320}}}, replace {{{-36n}}} with {{{-16n+-20n}}} (notice {{{-16n+-20n}}} adds up to {{{-36n}}}. So it is equivalent to {{{-36n}}})


{{{n^2+highlight(-16n+-20n)+320}}}



Now let's factor {{{n^2-16n-20n+320}}} by grouping:



{{{(n^2-16n)+(-20n+320)}}} Group like terms



{{{n(n-16)-20(n-16)}}} Factor out the GCF of {{{n}}} out of the first group. Factor out the GCF of {{{-20}}} out of the second group



{{{(n-20)(n-16)}}} Since we have a common term of {{{n-16}}}, we can combine like terms


So {{{n^2-16n-20n+320}}} factors to {{{(n-20)(n-16)}}}



So this also means that {{{n^2-36n+320}}} factors to {{{(n-20)(n-16)}}} (since {{{n^2-36n+320}}} is equivalent to {{{n^2-16n-20n+320}}})


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Answer:


So {{{n^2-36n+320}}} factors to {{{(n-20)(n-16)}}}