Question 190252


Looking at {{{x^2-9xy-10y^2}}} we can see that the first term is {{{x^2}}} and the last term is {{{-10y^2}}} where the coefficients are 1 and -10 respectively.


Now multiply the first coefficient 1 and the last coefficient -10 to get -10. Now what two numbers multiply to -10 and add to the  middle coefficient -9? Let's list all of the factors of -10:




Factors of -10:

1,2,5,10


-1,-2,-5,-10 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -10

(1)*(-10)

(2)*(-5)

(-1)*(10)

(-2)*(5)


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



Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-10</td><td>1+(-10)=-9</td></tr><tr><td align="center">2</td><td align="center">-5</td><td>2+(-5)=-3</td></tr><tr><td align="center">-1</td><td align="center">10</td><td>-1+10=9</td></tr><tr><td align="center">-2</td><td align="center">5</td><td>-2+5=3</td></tr></table>



From this list we can see that 1 and -10 add up to -9 and multiply to -10



Now looking at the expression {{{x^2-9xy-10y^2}}}, replace {{{-9xy}}} with {{{xy-10xy}}} (notice {{{xy-10xy}}} adds up to {{{-9xy}}}. So it is equivalent to {{{-9xy}}})


{{{x^2+highlight(xy-10xy)-10y^2}}}



Now let's factor {{{x^2+xy-10xy-10y^2}}} by grouping:



{{{(x^2+xy)+(-10xy-10y^2)}}} Group like terms



{{{x(x+y)-10y(x+y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-10y}}} out of the second group



{{{(x-10y)(x+y)}}} Since we have a common term of {{{x+y}}}, we can combine like terms


So {{{x^2+xy-10xy-10y^2}}} factors to {{{(x-10y)(x+y)}}}



So this also means that {{{x^2-9xy-10y^2}}} factors to {{{(x-10y)(x+y)}}} (since {{{x^2-9xy-10y^2}}} is equivalent to {{{x^2+xy-10xy-10y^2}}})




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     Answer:

So {{{x^2-9xy-10y^2}}} factors to {{{(x-10y)(x+y)}}}