Question 133000
{{{56 - 15w + w^2}}} Start with the given equation



{{{w^2- 15w+56}}} Rearrange the terms





Looking at {{{1w^2-15w+56}}} we can see that the first term is {{{1w^2}}} 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 the  middle coefficient -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 {{{1w^2-15w+56}}}, replace {{{-15w}}} with {{{-7w+-8w}}} (notice {{{-7w+-8w}}} adds up to {{{-15w}}}. So it is equivalent to {{{-15w}}})


{{{1w^2+highlight(-7w+-8w)+56}}}



Now let's factor {{{1w^2-7w-8w+56}}} by grouping:



{{{(1w^2-7w)+(-8w+56)}}} Group like terms



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



{{{(w-8)(w-7)}}} Since we have a common term of {{{w-7}}}, we can combine like terms


So {{{1w^2-7w-8w+56}}} factors to {{{(w-8)(w-7)}}}



So this also means that {{{1w^2-15w+56}}} factors to {{{(w-8)(w-7)}}} (since {{{1w^2-15w+56}}} is equivalent to {{{1w^2-7w-8w+56}}})




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     Answer:

So {{{w^2-15w+56}}} factors to {{{(w-8)(w-7)}}}