Question 131051


{{{6y^3+11y^2-35y}}} Start with the given expression



{{{y(6y^2+11y-35)}}} Factor out the GCF {{{y}}}



Now let's focus on the inner expression {{{6y^2+11y-35}}}





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Looking at {{{6y^2+11y-35}}} we can see that the first term is {{{6y^2}}} and the last term is {{{-35}}} where the coefficients are 6 and -35 respectively.


Now multiply the first coefficient 6 and the last coefficient -35 to get -210. Now what two numbers multiply to -210 and add to the  middle coefficient 11? Let's list all of the factors of -210:




Factors of -210:

1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210


-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -210

(1)*(-210)

(2)*(-105)

(3)*(-70)

(5)*(-42)

(6)*(-35)

(7)*(-30)

(10)*(-21)

(14)*(-15)

(-1)*(210)

(-2)*(105)

(-3)*(70)

(-5)*(42)

(-6)*(35)

(-7)*(30)

(-10)*(21)

(-14)*(15)


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



Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-210</td><td>1+(-210)=-209</td></tr><tr><td align="center">2</td><td align="center">-105</td><td>2+(-105)=-103</td></tr><tr><td align="center">3</td><td align="center">-70</td><td>3+(-70)=-67</td></tr><tr><td align="center">5</td><td align="center">-42</td><td>5+(-42)=-37</td></tr><tr><td align="center">6</td><td align="center">-35</td><td>6+(-35)=-29</td></tr><tr><td align="center">7</td><td align="center">-30</td><td>7+(-30)=-23</td></tr><tr><td align="center">10</td><td align="center">-21</td><td>10+(-21)=-11</td></tr><tr><td align="center">14</td><td align="center">-15</td><td>14+(-15)=-1</td></tr><tr><td align="center">-1</td><td align="center">210</td><td>-1+210=209</td></tr><tr><td align="center">-2</td><td align="center">105</td><td>-2+105=103</td></tr><tr><td align="center">-3</td><td align="center">70</td><td>-3+70=67</td></tr><tr><td align="center">-5</td><td align="center">42</td><td>-5+42=37</td></tr><tr><td align="center">-6</td><td align="center">35</td><td>-6+35=29</td></tr><tr><td align="center">-7</td><td align="center">30</td><td>-7+30=23</td></tr><tr><td align="center">-10</td><td align="center">21</td><td>-10+21=11</td></tr><tr><td align="center">-14</td><td align="center">15</td><td>-14+15=1</td></tr></table>



From this list we can see that -10 and 21 add up to 11 and multiply to -210



Now looking at the expression {{{6y^2+11y-35}}}, replace {{{11y}}} with {{{-10y+21y}}} (notice {{{-10y+21y}}} adds up to {{{11y}}}. So it is equivalent to {{{11y}}})


{{{6y^2+highlight(-10y+21y)+-35}}}



Now let's factor {{{6y^2-10y+21y-35}}} by grouping:



{{{(6y^2-10y)+(21y-35)}}} Group like terms



{{{2y(3y-5)+7(3y-5)}}} Factor out the GCF of {{{2y}}} out of the first group. Factor out the GCF of {{{7}}} out of the second group



{{{(2y+7)(3y-5)}}} Since we have a common term of {{{3y-5}}}, we can combine like terms


So {{{6y^2-10y+21y-35}}} factors to {{{(2y+7)(3y-5)}}}



So this also means that {{{6y^2+11y-35}}} factors to {{{(2y+7)(3y-5)}}} (since {{{6y^2+11y-35}}} is equivalent to {{{6y^2-10y+21y-35}}})




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So our expression goes from {{{y(6y^2+11y-35)}}} and factors further to {{{y(2y+7)(3y-5)}}}



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Answer:


So {{{6y^3+11y^2-35y}}} factors to {{{y(2y+7)(3y-5)}}}