Question 133704


Looking at {{{6n^2-19n+10}}} we can see that the first term is {{{6n^2}}} and the last term is {{{10}}} where the coefficients are 6 and 10 respectively.


Now multiply the first coefficient 6 and the last coefficient 10 to get 60. Now what two numbers multiply to 60 and add to the  middle coefficient -19? Let's list all of the factors of 60:




Factors of 60:

1,2,3,4,5,6,10,12,15,20,30,60


-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 60

1*60

2*30

3*20

4*15

5*12

6*10

(-1)*(-60)

(-2)*(-30)

(-3)*(-20)

(-4)*(-15)

(-5)*(-12)

(-6)*(-10)


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



Now which of these pairs add to -19? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -19


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">60</td><td>1+60=61</td></tr><tr><td align="center">2</td><td align="center">30</td><td>2+30=32</td></tr><tr><td align="center">3</td><td align="center">20</td><td>3+20=23</td></tr><tr><td align="center">4</td><td align="center">15</td><td>4+15=19</td></tr><tr><td align="center">5</td><td align="center">12</td><td>5+12=17</td></tr><tr><td align="center">6</td><td align="center">10</td><td>6+10=16</td></tr><tr><td align="center">-1</td><td align="center">-60</td><td>-1+(-60)=-61</td></tr><tr><td align="center">-2</td><td align="center">-30</td><td>-2+(-30)=-32</td></tr><tr><td align="center">-3</td><td align="center">-20</td><td>-3+(-20)=-23</td></tr><tr><td align="center">-4</td><td align="center">-15</td><td>-4+(-15)=-19</td></tr><tr><td align="center">-5</td><td align="center">-12</td><td>-5+(-12)=-17</td></tr><tr><td align="center">-6</td><td align="center">-10</td><td>-6+(-10)=-16</td></tr></table>



From this list we can see that -4 and -15 add up to -19 and multiply to 60



Now looking at the expression {{{6n^2-19n+10}}}, replace {{{-19n}}} with {{{-4n+-15n}}} (notice {{{-4n+-15n}}} adds up to {{{-19n}}}. So it is equivalent to {{{-19n}}})


{{{6n^2+highlight(-4n+-15n)+10}}}



Now let's factor {{{6n^2-4n-15n+10}}} by grouping:



{{{(6n^2-4n)+(-15n+10)}}} Group like terms



{{{2n(3n-2)-5(3n-2)}}} Factor out the GCF of {{{2n}}} out of the first group. Factor out the GCF of {{{-5}}} out of the second group



{{{(2n-5)(3n-2)}}} Since we have a common term of {{{3n-2}}}, we can combine like terms


So {{{6n^2-4n-15n+10}}} factors to {{{(2n-5)(3n-2)}}}



So this also means that {{{6n^2-19n+10}}} factors to {{{(2n-5)(3n-2)}}} (since {{{6n^2-19n+10}}} is equivalent to {{{6n^2-4n-15n+10}}})




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     Answer:

So {{{6n^2-19n+10}}} factors to {{{(2n-5)(3n-2)}}}