Question 117565
#1




{{{r^3+7r^2-18r}}} Start with the given expression



{{{r(r^2+7r-18)}}} Factor out the GCF {{{r}}}



Now let's focus on the inner expression {{{r^2+7r-18}}}




Looking at {{{r^2+7r-18}}} we can see that the first term is {{{r^2}}} and the last term is {{{-18}}} where the coefficients are 1 and -18 respectively.


Now multiply the first coefficient 1 and the last coefficient -18 to get -18. Now what two numbers multiply to -18 and add to the  middle coefficient 7? Let's list all of the factors of -18:




Factors of -18:

1,2,3,6,9,18


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


These factors pair up and multiply to -18

(1)*(-18)

(2)*(-9)

(3)*(-6)

(-1)*(18)

(-2)*(9)

(-3)*(6)


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



Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-18</td><td>1+(-18)=-17</td></tr><tr><td align="center">2</td><td align="center">-9</td><td>2+(-9)=-7</td></tr><tr><td align="center">3</td><td align="center">-6</td><td>3+(-6)=-3</td></tr><tr><td align="center">-1</td><td align="center">18</td><td>-1+18=17</td></tr><tr><td align="center">-2</td><td align="center">9</td><td>-2+9=7</td></tr><tr><td align="center">-3</td><td align="center">6</td><td>-3+6=3</td></tr></table>



From this list we can see that -2 and 9 add up to 7 and multiply to -18



Now looking at the expression {{{r^2+7r-18}}}, replace {{{7r}}} with {{{-2r+9r}}} (notice {{{-2r+9r}}} adds up to {{{7r}}}. So it is equivalent to {{{7r}}})


{{{r^2+highlight(-2r+9r)+-18}}}



Now let's factor {{{r^2-2r+9r-18}}} by grouping:



{{{(r^2-2r)+(9r-18)}}} Group like terms



{{{r(r-2)+9(r-2)}}} Factor out the GCF of {{{r}}} out of the first group. Factor out the GCF of {{{9}}} out of the second group



{{{(r+9)(r-2)}}} Since we have a common term of {{{r-2}}}, we can combine like terms




So {{{r^2+7r-18}}} factors to {{{(r+9)(r-2)}}}



{{{r(r+9)(r-2)}}} Now reintroduce the GCF {{{r}}}


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Answer:


So {{{r^3+7r^2-18r}}} factors to {{{r(r+9)(r-2)}}}



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#2 





Looking at {{{x^2y-9xy^2-36y^3}}} we can see that the first term is {{{x^2y}}} and the last term is {{{-36y^3}}} where the coefficients are 1 and -36 respectively.


Now multiply the first coefficient 1 and the last coefficient -36 to get -36. Now what two numbers multiply to -36 and add to the  middle coefficient -9? Let's list all of the factors of -36:




Factors of -36:

1,2,3,4,6,9,12,18


-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -36

(1)*(-36)

(2)*(-18)

(3)*(-12)

(4)*(-9)

(-1)*(36)

(-2)*(18)

(-3)*(12)

(-4)*(9)


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



Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-36</td><td>1+(-36)=-35</td></tr><tr><td align="center">2</td><td align="center">-18</td><td>2+(-18)=-16</td></tr><tr><td align="center">3</td><td align="center">-12</td><td>3+(-12)=-9</td></tr><tr><td align="center">4</td><td align="center">-9</td><td>4+(-9)=-5</td></tr><tr><td align="center">-1</td><td align="center">36</td><td>-1+36=35</td></tr><tr><td align="center">-2</td><td align="center">18</td><td>-2+18=16</td></tr><tr><td align="center">-3</td><td align="center">12</td><td>-3+12=9</td></tr><tr><td align="center">-4</td><td align="center">9</td><td>-4+9=5</td></tr></table>



From this list we can see that 3 and -12 add up to -9 and multiply to -36



Now looking at the expression {{{x^2y-9xy^2-36y^3}}}, replace {{{-9xy^2}}} with {{{3xy^2+-12xy^2}}} (notice {{{3xy^2+-12xy^2}}} adds up to {{{-9xy^2}}}. So it is equivalent to {{{-9xy^2}}})


{{{x^2y+highlight(3xy^2+-12xy^2)+-36y^3}}}



Now let's factor {{{x^2y+3xy^2-12xy^2-36y^3}}} by grouping:



{{{(x^2y+3xy^2)+(-12xy^2-36y^3)}}} Group like terms



{{{xy(x+3y)-12y^2(x+3y)}}} Factor out the GCF of {{{xy}}} out of the first group. Factor out the GCF of {{{-12y^2}}} out of the second group



{{{(xy-12y^2)(x+3y)}}} Since we have a common term of {{{x+3y}}}, we can combine like terms


So {{{x^2y+3xy^2-12xy^2-36y^3}}} factors to {{{(xy-12y^2)(x+3y)}}}



So this also means that {{{x^2y-9xy^2-36y^3}}} factors to {{{(xy-12y^2)(x+3y)}}} (since {{{x^2y-9xy^2-36y^3}}} is equivalent to {{{x^2y+3xy^2-12xy^2-36y^3}}})


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Answer:


So {{{x^2y-9xy^2-36y^3}}} factors to {{{(xy-12y^2)(x+3y)}}}