Question 120310

{{{9bn^3+15bn^2-14bn}}} Start with the given expression



{{{bn(9n^2+15n-14)}}} Factor out the GCF {{{bn}}}



Now let's focus on the inner expression {{{9n^2+15n-14}}}





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Looking at {{{9n^2+15n-14}}} we can see that the first term is {{{9n^2}}} and the last term is {{{-14}}} where the coefficients are 9 and -14 respectively.


Now multiply the first coefficient 9 and the last coefficient -14 to get -126. Now what two numbers multiply to -126 and add to the  middle coefficient 15? Let's list all of the factors of -126:




Factors of -126:

1,2,3,6,7,9,14,18,21,42,63,126


-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -126

(1)*(-126)

(2)*(-63)

(3)*(-42)

(6)*(-21)

(7)*(-18)

(9)*(-14)

(-1)*(126)

(-2)*(63)

(-3)*(42)

(-6)*(21)

(-7)*(18)

(-9)*(14)


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



Now which of these pairs add to 15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 15


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-126</td><td>1+(-126)=-125</td></tr><tr><td align="center">2</td><td align="center">-63</td><td>2+(-63)=-61</td></tr><tr><td align="center">3</td><td align="center">-42</td><td>3+(-42)=-39</td></tr><tr><td align="center">6</td><td align="center">-21</td><td>6+(-21)=-15</td></tr><tr><td align="center">7</td><td align="center">-18</td><td>7+(-18)=-11</td></tr><tr><td align="center">9</td><td align="center">-14</td><td>9+(-14)=-5</td></tr><tr><td align="center">-1</td><td align="center">126</td><td>-1+126=125</td></tr><tr><td align="center">-2</td><td align="center">63</td><td>-2+63=61</td></tr><tr><td align="center">-3</td><td align="center">42</td><td>-3+42=39</td></tr><tr><td align="center">-6</td><td align="center">21</td><td>-6+21=15</td></tr><tr><td align="center">-7</td><td align="center">18</td><td>-7+18=11</td></tr><tr><td align="center">-9</td><td align="center">14</td><td>-9+14=5</td></tr></table>



From this list we can see that -6 and 21 add up to 15 and multiply to -126



Now looking at the expression {{{9n^2+15n-14}}}, replace {{{15n}}} with {{{-6n+21n}}} (notice {{{-6n+21n}}} adds up to {{{15n}}}. So it is equivalent to {{{15n}}})


{{{9n^2+highlight(-6n+21n)+-14}}}



Now let's factor {{{9n^2-6n+21n-14}}} by grouping:



{{{(9n^2-6n)+(21n-14)}}} Group like terms



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



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


So {{{9n^2-6n+21n-14}}} factors to {{{(3n+7)(3n-2)}}}



So this also means that {{{9n^2+15n-14}}} factors to {{{(3n+7)(3n-2)}}} (since {{{9n^2+15n-14}}} is equivalent to {{{9n^2-6n+21n-14}}})




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So our expression goes from {{{bn(9n^2+15n-14)}}} and factors further to {{{bn(3n+7)(3n-2)}}}



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Answer:


So {{{9bn^3+15bn^2-14bn}}} factors to {{{bn(3n+7)(3n-2)}}}