Question 111328
Looking at {{{a^4-15a^2+56}}} we can see that the first term is {{{a^4}}} and the last term is {{{56}}} where the coefficients are 1 and 56 respectively.


Now multiply the first coefficient 1 and the last coefficient 56 to get 56. Now what two numbers multiply to 56 and add to -15? Let's list all of the factors of 56:




Factors of 56:

1,2,4,7,8,14,28,56


-1,-2,-4,-7,-8,-14,-28,-56 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 56

1*56

2*28

4*14

7*8

(-1)*(-56)

(-2)*(-28)

(-4)*(-14)

(-7)*(-8)


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



Now which of these pairs add to -15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -15


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">56</td><td>1+56=57</td></tr><tr><td align="center">2</td><td align="center">28</td><td>2+28=30</td></tr><tr><td align="center">4</td><td align="center">14</td><td>4+14=18</td></tr><tr><td align="center">7</td><td align="center">8</td><td>7+8=15</td></tr><tr><td align="center">-1</td><td align="center">-56</td><td>-1+(-56)=-57</td></tr><tr><td align="center">-2</td><td align="center">-28</td><td>-2+(-28)=-30</td></tr><tr><td align="center">-4</td><td align="center">-14</td><td>-4+(-14)=-18</td></tr><tr><td align="center">-7</td><td align="center">-8</td><td>-7+(-8)=-15</td></tr></table>



From this list we can see that -7 and -8 add up to -15 and multiply to 56



Now looking at the expression {{{a^4-15a^2+56}}}, replace {{{-15a^2}}} with {{{-7a^2+-8a^2}}} (notice {{{-7a^2+-8a^2}}} adds up to {{{-15a^2}}}. So it is equivalent to {{{-15a^2}}})


{{{a^4+highlight(-7a^2+-8a^2)+56}}}



Now let's factor {{{a^4-7a^2-8a^2+56}}} by grouping:



{{{(a^4-7a^2)+(-8a^2+56)}}} Group like terms



{{{a^2(a^2-7)-8(a^2-7)}}} Factor out the GCF of {{{a^2}}} out of the first group. Factor out the GCF of {{{8}}} out of the second group



So this also means that {{{a^4-15a^2+56}}} factors to {{{(a^2-8)(a^2-7)}}} (since {{{a^4-15a^2+56}}} is equivalent to {{{a^4-7a^2-8a^2+56}}})