Question 116200


Looking at {{{x^2-11xy-42y^2}}} we can see that the first term is {{{x^2}}} and the last term is {{{-42y^2}}} where the coefficients are 1 and -42 respectively.


Now multiply the first coefficient 1 and the last coefficient -42 to get -42. Now what two numbers multiply to -42 and add to the  middle coefficient -11? Let's list all of the factors of -42:




Factors of -42:

1,2,3,6,7,14,21,42


-1,-2,-3,-6,-7,-14,-21,-42 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -42

(1)*(-42)

(2)*(-21)

(3)*(-14)

(6)*(-7)

(-1)*(42)

(-2)*(21)

(-3)*(14)

(-6)*(7)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -11


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-42</td><td>1+(-42)=-41</td></tr><tr><td align="center">2</td><td align="center">-21</td><td>2+(-21)=-19</td></tr><tr><td align="center">3</td><td align="center">-14</td><td>3+(-14)=-11</td></tr><tr><td align="center">6</td><td align="center">-7</td><td>6+(-7)=-1</td></tr><tr><td align="center">-1</td><td align="center">42</td><td>-1+42=41</td></tr><tr><td align="center">-2</td><td align="center">21</td><td>-2+21=19</td></tr><tr><td align="center">-3</td><td align="center">14</td><td>-3+14=11</td></tr><tr><td align="center">-6</td><td align="center">7</td><td>-6+7=1</td></tr></table>



From this list we can see that 3 and -14 add up to -11 and multiply to -42



Now looking at the expression {{{x^2-11xy-42y^2}}}, replace {{{-11xy}}} with {{{3xy+-14xy}}} (notice {{{3xy+-14xy}}} adds up to {{{-11xy}}}. So it is equivalent to {{{-11xy}}})


{{{x^2+highlight(3xy+-14xy)+-42y^2}}}



Now let's factor {{{x^2+3xy-14xy-42y^2}}} by grouping:



{{{(x^2+3xy)+(-14xy-42y^2)}}} Group like terms



{{{x(x+3y)-14y(x+3y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-14y}}} out of the second group



{{{(x-14y)(x+3y)}}} Since we have a common term of {{{x+3y}}}, we can combine like terms


So {{{x^2+3xy-14xy-42y^2}}} factors to {{{(x-14y)(x+3y)}}}



So this also means that {{{x^2-11xy-42y^2}}} factors to {{{(x-14y)(x+3y)}}} (since {{{x^2-11xy-42y^2}}} is equivalent to {{{x^2+3xy-14xy-42y^2}}})


-------------------------------

Answer:


So {{{x^2-11xy-42y^2}}} factors to {{{(x-14y)(x+3y)}}}