Question 132328


{{{5y^9-53y^6+72y^3}}} Start with the given expression



{{{y^3(5y^6-53y^3+72)}}} Factor out the GCF {{{y^3}}}



Now let's focus on the inner expression {{{5y^6-53y^3+72}}}





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Looking at {{{5y^6-53y^3+72}}} we can see that the first term is {{{5y^6}}} and the last term is {{{72}}} where the coefficients are 5 and 72 respectively.


Now multiply the first coefficient 5 and the last coefficient 72 to get 360. Now what two numbers multiply to 360 and add to the  middle coefficient -53? Let's list all of the factors of 360:




Factors of 360:

1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360


-1,-2,-3,-4,-5,-6,-8,-9,-10,-12,-15,-18,-20,-24,-30,-36,-40,-45,-60,-72,-90,-120,-180,-360 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 360

1*360

2*180

3*120

4*90

5*72

6*60

8*45

9*40

10*36

12*30

15*24

18*20

(-1)*(-360)

(-2)*(-180)

(-3)*(-120)

(-4)*(-90)

(-5)*(-72)

(-6)*(-60)

(-8)*(-45)

(-9)*(-40)

(-10)*(-36)

(-12)*(-30)

(-15)*(-24)

(-18)*(-20)


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



Now which of these pairs add to -53? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -53


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">360</td><td>1+360=361</td></tr><tr><td align="center">2</td><td align="center">180</td><td>2+180=182</td></tr><tr><td align="center">3</td><td align="center">120</td><td>3+120=123</td></tr><tr><td align="center">4</td><td align="center">90</td><td>4+90=94</td></tr><tr><td align="center">5</td><td align="center">72</td><td>5+72=77</td></tr><tr><td align="center">6</td><td align="center">60</td><td>6+60=66</td></tr><tr><td align="center">8</td><td align="center">45</td><td>8+45=53</td></tr><tr><td align="center">9</td><td align="center">40</td><td>9+40=49</td></tr><tr><td align="center">10</td><td align="center">36</td><td>10+36=46</td></tr><tr><td align="center">12</td><td align="center">30</td><td>12+30=42</td></tr><tr><td align="center">15</td><td align="center">24</td><td>15+24=39</td></tr><tr><td align="center">18</td><td align="center">20</td><td>18+20=38</td></tr><tr><td align="center">-1</td><td align="center">-360</td><td>-1+(-360)=-361</td></tr><tr><td align="center">-2</td><td align="center">-180</td><td>-2+(-180)=-182</td></tr><tr><td align="center">-3</td><td align="center">-120</td><td>-3+(-120)=-123</td></tr><tr><td align="center">-4</td><td align="center">-90</td><td>-4+(-90)=-94</td></tr><tr><td align="center">-5</td><td align="center">-72</td><td>-5+(-72)=-77</td></tr><tr><td align="center">-6</td><td align="center">-60</td><td>-6+(-60)=-66</td></tr><tr><td align="center">-8</td><td align="center">-45</td><td>-8+(-45)=-53</td></tr><tr><td align="center">-9</td><td align="center">-40</td><td>-9+(-40)=-49</td></tr><tr><td align="center">-10</td><td align="center">-36</td><td>-10+(-36)=-46</td></tr><tr><td align="center">-12</td><td align="center">-30</td><td>-12+(-30)=-42</td></tr><tr><td align="center">-15</td><td align="center">-24</td><td>-15+(-24)=-39</td></tr><tr><td align="center">-18</td><td align="center">-20</td><td>-18+(-20)=-38</td></tr></table>



From this list we can see that -8 and -45 add up to -53 and multiply to 360



Now looking at the expression {{{5y^6-53y^3+72}}}, replace {{{-53y^3}}} with {{{-8y^3+-45y^3}}} (notice {{{-8y^3+-45y^3}}} adds up to {{{-53y^3}}}. So it is equivalent to {{{-53y^3}}})


{{{5y^6+highlight(-8y^3+-45y^3)+72}}}



Now let's factor {{{5y^6-8y^3-45y^3+72}}} by grouping:



{{{(5y^6-8y^3)+(-45y^3+72)}}} Group like terms



{{{y^3(5y^3-8)-9(5y^3-8)}}} Factor out the GCF of {{{y^3}}} out of the first group. Factor out the GCF of {{{-9}}} out of the second group



{{{(y^3-9)(5y^3-8)}}} Since we have a common term of {{{5y^3-8}}}, we can combine like terms


So {{{5y^6-8y^3-45y^3+72}}} factors to {{{(y^3-9)(5y^3-8)}}}



So this also means that {{{5y^6-53y^3+72}}} factors to {{{(y^3-9)(5y^3-8)}}} (since {{{5y^6-53y^3+72}}} is equivalent to {{{5y^6-8y^3-45y^3+72}}})




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So our expression goes from {{{y^3(5y^6-53y^3+72)}}} and factors further to {{{y^3(y^3-9)(5y^3-8)}}}



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Answer:


So {{{5y^9-53y^6+72y^3}}} factors to {{{y^3(y^3-9)(5y^3-8)}}}