Question 221874
# 1



{{{27y^5-63y^4+30y^3}}} Start with the given expression



{{{3y^3(9y^2-21y+10)}}} Factor out the GCF {{{3y^3}}}



Now let's focus on the inner expression {{{9y^2-21y+10}}}





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Looking at {{{9y^2-21y+10}}} we can see that the first term is {{{9y^2}}} and the last term is {{{10}}} where the coefficients are 9 and 10 respectively.


Now multiply the first coefficient 9 and the last coefficient 10 to get 90. Now what two numbers multiply to 90 and add to the  middle coefficient -21? Let's list all of the factors of 90:




Factors of 90:

1,2,3,5,6,9,10,15,18,30,45,90


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


These factors pair up and multiply to 90

1*90

2*45

3*30

5*18

6*15

9*10

(-1)*(-90)

(-2)*(-45)

(-3)*(-30)

(-5)*(-18)

(-6)*(-15)

(-9)*(-10)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -21? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -21


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">90</td><td>1+90=91</td></tr><tr><td align="center">2</td><td align="center">45</td><td>2+45=47</td></tr><tr><td align="center">3</td><td align="center">30</td><td>3+30=33</td></tr><tr><td align="center">5</td><td align="center">18</td><td>5+18=23</td></tr><tr><td align="center">6</td><td align="center">15</td><td>6+15=21</td></tr><tr><td align="center">9</td><td align="center">10</td><td>9+10=19</td></tr><tr><td align="center">-1</td><td align="center">-90</td><td>-1+(-90)=-91</td></tr><tr><td align="center">-2</td><td align="center">-45</td><td>-2+(-45)=-47</td></tr><tr><td align="center">-3</td><td align="center">-30</td><td>-3+(-30)=-33</td></tr><tr><td align="center">-5</td><td align="center">-18</td><td>-5+(-18)=-23</td></tr><tr><td align="center">-6</td><td align="center">-15</td><td>-6+(-15)=-21</td></tr><tr><td align="center">-9</td><td align="center">-10</td><td>-9+(-10)=-19</td></tr></table>



From this list we can see that -6 and -15 add up to -21 and multiply to 90



Now looking at the expression {{{9y^2-21y+10}}}, replace {{{-21y}}} with {{{-6y+-15y}}} (notice {{{-6y+-15y}}} adds up to {{{-21y}}}. So it is equivalent to {{{-21y}}})


{{{9y^2+highlight(-6y+-15y)+10}}}



Now let's factor {{{9y^2-6y-15y+10}}} by grouping:



{{{(9y^2-6y)+(-15y+10)}}} Group like terms



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



{{{(3y-5)(3y-2)}}} Since we have a common term of {{{3y-2}}}, we can combine like terms


So {{{9y^2-6y-15y+10}}} factors to {{{(3y-5)(3y-2)}}}



So this also means that {{{9y^2-21y+10}}} factors to {{{(3y-5)(3y-2)}}} (since {{{9y^2-21y+10}}} is equivalent to {{{9y^2-6y-15y+10}}})




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So our expression goes from {{{3y^3(9y^2-21y+10)}}} and factors further to {{{3y^3(3y-5)(3y-2)}}}



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Answer:


So {{{27y^5-63y^4+30y^3}}} completely factors to {{{3y^3(3y-5)(3y-2)}}}



In other words, {{{27y^5-63y^4+30y^3=3y^3(3y-5)(3y-2)}}}

    

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# 2




{{{125-8u^3}}} Start with the given expression.



{{{(5)^3-(2u)^3}}} Rewrite {{{125}}} as {{{(5)^3}}}. Rewrite {{{8u^3}}} as {{{(2u)^3}}}.



{{{(5-2u)((5)^2+(5)(2u)+(2u)^2)}}} Now factor by using the difference of cubes formula. Remember the <a href="http://www.purplemath.com/modules/specfact2.htm">difference of cubes formula</a> is {{{A^3-B^3=(A-B)(A^2+AB+B^2)}}}



{{{(5-2u)(25+10u+4u^2)}}} Multiply


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Answer:

So {{{125-8u^3}}} factors to {{{(5-2u)(25+10u+4u^2)}}}.


In other words, {{{125-8u^3=(5-2u)(25+10u+4u^2)}}}