Question 117342

{{{x^3-26x^2+48x}}} Start with the given expression



{{{x(x^2-26x+48)}}} Factor out the GCF {{{x}}}



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





Looking at {{{x^2-26x+48}}} we can see that the first term is {{{x^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 -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 {{{x^2-26x+48}}}, replace {{{-26x}}} with {{{-2x+-24x}}} (notice {{{-2x+-24x}}} adds up to {{{-26x}}}. So it is equivalent to {{{-26x}}})


{{{x^2+highlight(-2x+-24x)+48}}}



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



{{{(x^2-2x)+(-24x+48)}}} Group like terms



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



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




So {{{x^2-26x+48}}} factors to {{{(x-24)(x-2)}}}



{{{x(x-24)(x-2)}}} Reintroduce the GCF



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Answer:


So {{{x^3-26x^2+48x}}}  factors to {{{x(x-24)(x-2)}}}