Question 156341
{{{18xy^3+3xy^2-10xy}}} Start with the given expression



{{{xy(18y^2+3y-10)}}} Factor out the GCF {{{xy}}}



Now let's focus on the inner expression {{{18xy^2+3y-10}}}



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Looking at {{{18y^2+3y-10}}} we can see that the first term is {{{18y^2}}} and the last term is {{{-10}}} where the coefficients are 18 and -10 respectively.


Now multiply the first coefficient 18 and the last coefficient -10 to get -180. Now what two numbers multiply to -180 and add to the  middle coefficient 3? Let's list all of the factors of -180:




Factors of -180:

1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180


-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-30,-36,-45,-60,-90,-180 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -180

(1)*(-180)

(2)*(-90)

(3)*(-60)

(4)*(-45)

(5)*(-36)

(6)*(-30)

(9)*(-20)

(10)*(-18)

(12)*(-15)

(-1)*(180)

(-2)*(90)

(-3)*(60)

(-4)*(45)

(-5)*(36)

(-6)*(30)

(-9)*(20)

(-10)*(18)

(-12)*(15)


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



Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-180</td><td>1+(-180)=-179</td></tr><tr><td align="center">2</td><td align="center">-90</td><td>2+(-90)=-88</td></tr><tr><td align="center">3</td><td align="center">-60</td><td>3+(-60)=-57</td></tr><tr><td align="center">4</td><td align="center">-45</td><td>4+(-45)=-41</td></tr><tr><td align="center">5</td><td align="center">-36</td><td>5+(-36)=-31</td></tr><tr><td align="center">6</td><td align="center">-30</td><td>6+(-30)=-24</td></tr><tr><td align="center">9</td><td align="center">-20</td><td>9+(-20)=-11</td></tr><tr><td align="center">10</td><td align="center">-18</td><td>10+(-18)=-8</td></tr><tr><td align="center">12</td><td align="center">-15</td><td>12+(-15)=-3</td></tr><tr><td align="center">-1</td><td align="center">180</td><td>-1+180=179</td></tr><tr><td align="center">-2</td><td align="center">90</td><td>-2+90=88</td></tr><tr><td align="center">-3</td><td align="center">60</td><td>-3+60=57</td></tr><tr><td align="center">-4</td><td align="center">45</td><td>-4+45=41</td></tr><tr><td align="center">-5</td><td align="center">36</td><td>-5+36=31</td></tr><tr><td align="center">-6</td><td align="center">30</td><td>-6+30=24</td></tr><tr><td align="center">-9</td><td align="center">20</td><td>-9+20=11</td></tr><tr><td align="center">-10</td><td align="center">18</td><td>-10+18=8</td></tr><tr><td align="center"><font color="red">-12</font></td><td align="center"><font color="red">15</font></td><td><font color="red">-12+15=3</font></td></tr></table>



From this list we can see that -12 and 15 add up to 3 and multiply to -180



Now looking at the expression {{{18xy^2+3y-10}}}, replace {{{3y}}} with {{{-12y+15y}}} (Remember, if {{{-12+15=3}}}, then {{{-12y+15y=3y}}})



{{{18y^2+highlight(-12y+15y)+-10}}}



Now let's factor {{{18y^2-12y+15y-10}}} by grouping:



{{{(18y^2-12y)+(15y-10)}}} Group like terms



{{{6y(3y-2)+5(3y-2)}}} Factor out the GCF of {{{6y}}} out of the first group. Factor out the GCF of {{{5}}} out of the second group



So this also means that {{{18y^2+3y-10}}} factors to {{{(6y+5)(3y-2)}}} (since {{{18y^2+3y-10}}} is equivalent to {{{18y^2-12y+15y-10}}})




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So the expression {{{xy(18y^2+3y-10)}}} factors further to {{{xy(6y+5)(3y-2)}}}




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Answer:



So {{{18xy^3+3xy^2-10xy}}} completely factors to {{{xy(6y+5)(3y-2)}}}



To check the answer, simply FOIL and expand {{{xy(6y+5)(3y-2)}}} to get {{{18xy^3+3xy^2-10xy}}}