Question 184675
{{{x^3-60xy^2+7x^2y}}} Start with the given expression



{{{x^3+7x^2y-60xy^2}}} Rearrange the terms.



{{{x(x^2+7xy-60y^2)}}} Factor out the GCF {{{x}}}



Now let's focus on the inner expression {{{x^2+7xy-60y^2}}}





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Looking at {{{1x^2+7xy-60y^2}}} we can see that the first term is {{{1x^2}}} and the last term is {{{-60y^2}}} where the coefficients are 1 and -60 respectively.


Now multiply the first coefficient 1 and the last coefficient -60 to get -60. Now what two numbers multiply to -60 and add to the  middle coefficient 7? 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, the product of a negative and a positive number is a negative number



Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7


<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)=-59</td></tr><tr><td align="center">2</td><td align="center">-30</td><td>2+(-30)=-28</td></tr><tr><td align="center">3</td><td align="center">-20</td><td>3+(-20)=-17</td></tr><tr><td align="center">4</td><td align="center">-15</td><td>4+(-15)=-11</td></tr><tr><td align="center">5</td><td align="center">-12</td><td>5+(-12)=-7</td></tr><tr><td align="center">6</td><td align="center">-10</td><td>6+(-10)=-4</td></tr><tr><td align="center">-1</td><td align="center">60</td><td>-1+60=59</td></tr><tr><td align="center">-2</td><td align="center">30</td><td>-2+30=28</td></tr><tr><td align="center">-3</td><td align="center">20</td><td>-3+20=17</td></tr><tr><td align="center">-4</td><td align="center">15</td><td>-4+15=11</td></tr><tr><td align="center">-5</td><td align="center">12</td><td>-5+12=7</td></tr><tr><td align="center">-6</td><td align="center">10</td><td>-6+10=4</td></tr></table>



From this list we can see that -5 and 12 add up to 7 and multiply to -60



Now looking at the expression {{{1x^2+7xy-60y^2}}}, replace {{{7xy}}} with {{{-5xy+12xy}}} (notice {{{-5xy+12xy}}} adds up to {{{7xy}}}. So it is equivalent to {{{7xy}}})


{{{1x^2+highlight(-5xy+12xy)+-60y^2}}}



Now let's factor {{{1x^2-5xy+12xy-60y^2}}} by grouping:



{{{(1x^2-5xy)+(12xy-60y^2)}}} Group like terms



{{{x(x-5y)+12y(x-5y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{12y}}} out of the second group



{{{(x+12y)(x-5y)}}} Since we have a common term of {{{x-5y}}}, we can combine like terms


So {{{1x^2-5xy+12xy-60y^2}}} factors to {{{(x+12y)(x-5y)}}}



So this also means that {{{1x^2+7xy-60y^2}}} factors to {{{(x+12y)(x-5y)}}} (since {{{1x^2+7xy-60y^2}}} is equivalent to {{{1x^2-5xy+12xy-60y^2}}})




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So our expression goes from {{{x(x^2+7xy-60y^2)}}} and factors further to {{{x(x+12y)(x-5y)}}}



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Answer:


So {{{x^3-60xy^2+7x^2y}}} factors to {{{x(x+12y)(x-5y)}}}