Question 197362

{{{r^3-2r^2-48r}}} Start with the given expression



{{{r(r^2-2r-48)}}} Factor out the GCF {{{r}}}



Now let's focus on the inner expression {{{r^2-2r-48}}}





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Looking at {{{r^2-2r-48}}} we can see that the first term is {{{r^2}}} and the last term is {{{-48}}} where the coefficients are 1 and -48 respectively.


Now multiply the first coefficient 1 and the last coefficient -48 to get -48. Now what two numbers multiply to -48 and add to the  middle coefficient -2? Let's list all of the factors of -48:




Factors of -48:

1,2,3,4,6,8,12,16,24,48


-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -48

(1)*(-48)

(2)*(-24)

(3)*(-16)

(4)*(-12)

(6)*(-8)

(-1)*(48)

(-2)*(24)

(-3)*(16)

(-4)*(12)

(-6)*(8)


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



Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-48</td><td>1+(-48)=-47</td></tr><tr><td align="center">2</td><td align="center">-24</td><td>2+(-24)=-22</td></tr><tr><td align="center">3</td><td align="center">-16</td><td>3+(-16)=-13</td></tr><tr><td align="center">4</td><td align="center">-12</td><td>4+(-12)=-8</td></tr><tr><td align="center">6</td><td align="center">-8</td><td>6+(-8)=-2</td></tr><tr><td align="center">-1</td><td align="center">48</td><td>-1+48=47</td></tr><tr><td align="center">-2</td><td align="center">24</td><td>-2+24=22</td></tr><tr><td align="center">-3</td><td align="center">16</td><td>-3+16=13</td></tr><tr><td align="center">-4</td><td align="center">12</td><td>-4+12=8</td></tr><tr><td align="center">-6</td><td align="center">8</td><td>-6+8=2</td></tr></table>



From this list we can see that 6 and -8 add up to -2 and multiply to -48



Now looking at the expression {{{r^2-2r-48}}}, replace {{{-2r}}} with {{{6r-8r}}} (notice {{{6r-8r}}} combines back to {{{-2r}}}. So it is equivalent to {{{-2r}}})


{{{r^2+highlight(6r-8r)-48}}}



Now let's factor {{{r^2+6r-8r-48}}} by grouping:



{{{(r^2+6r)+(-8r-48)}}} Group like terms



{{{r(r+6)-8(r+6)}}} Factor out the GCF of {{{r}}} out of the first group. Factor out the GCF of {{{-8}}} out of the second group



{{{(r-8)(r+6)}}} Since we have a common term of {{{r+6}}}, we can combine like terms


So {{{r^2+6r-8r-48}}} factors to {{{(r-8)(r+6)}}}



So this also means that {{{r^2-2r-48}}} factors to {{{(r-8)(r+6)}}} (since {{{r^2-2r-48}}} is equivalent to {{{r^2+6r-8r-48}}})




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So our expression goes from {{{r(r^2-2r-48)}}} and factors further to {{{r(r-8)(r+6)}}}



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Answer:


So {{{r^3-2r^2-48r}}} completely factors to {{{r(r-8)(r+6)}}}