Question 114213

{{{2a^3-52a^2b+96ab^2}}} Start with the given expression



{{{2a(a^2-26ab+48b^2)}}} Factor out the GCF {{{2a}}}



Now let's focus on the inner expression {{{a^2-26ab+48b^2}}}


Looking at {{{a^2-26ab+48b^2}}} we can see that the first term is {{{a^2}}} and the last term is {{{48b^2}}} 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 -26? 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 two negative numbers multiplied together make a positive number



Now which of these pairs add to -26? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -26


<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=49</td></tr><tr><td align="center">2</td><td align="center">24</td><td>2+24=26</td></tr><tr><td align="center">3</td><td align="center">16</td><td>3+16=19</td></tr><tr><td align="center">4</td><td align="center">12</td><td>4+12=16</td></tr><tr><td align="center">6</td><td align="center">8</td><td>6+8=14</td></tr><tr><td align="center">-1</td><td align="center">-48</td><td>-1+(-48)=-49</td></tr><tr><td align="center">-2</td><td align="center">-24</td><td>-2+(-24)=-26</td></tr><tr><td align="center">-3</td><td align="center">-16</td><td>-3+(-16)=-19</td></tr><tr><td align="center">-4</td><td align="center">-12</td><td>-4+(-12)=-16</td></tr><tr><td align="center">-6</td><td align="center">-8</td><td>-6+(-8)=-14</td></tr></table>



From this list we can see that -2 and -24 add up to -26 and multiply to 48



Now looking at the expression {{{a^2-26ab+48b^2}}}, replace {{{-26ab}}} with {{{-2ab+-24ab}}} (notice {{{-2ab+-24ab}}} adds up to {{{-26ab}}}. So it is equivalent to {{{-26ab}}})


{{{a^2+highlight(-2ab+-24ab)+48b^2}}}



Now let's factor {{{a^2-2ab-24ab+48b^2}}} by grouping:



{{{(a^2-2ab)+(-24ab+48b^2)}}} Group like terms



{{{a(a-2b)-24b(a-2b)}}} Factor out the GCF of {{{a}}} out of the first group. Factor out the GCF of {{{-24b}}} out of the second group



{{{(a-24b)(a-2b)}}} Since we have a common term of {{{a-2b}}}, we can combine like terms


So {{{a^2-2ab-24ab+48b^2}}} factors to {{{(a-24b)(a-2b)}}}



So this also means that {{{a^2-26ab+48b^2}}} factors to {{{(a-24b)(a-2b)}}} (since {{{a^2-26ab+48b^2}}} is equivalent to {{{a^2-2ab-24ab+48b^2}}})



{{{2a(a-24b)(a-2b)}}}Now reintroduce the GCF back in



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Answer:


So {{{2a^3-52a^2b+96ab^2}}} factors to {{{2a(a-24b)(a-2b)}}}