Question 407841


Looking at {{{3x^2-5x-884}}} we can see that the first term is {{{3x^2}}} and the last term is {{{-884}}} where the coefficients are 3 and -884 respectively.


Now multiply the first coefficient 3 and the last coefficient -884 to get -2652. Now what two numbers multiply to -2652 and add to the  middle coefficient -5? Let's list all of the factors of -2652:




Factors of -2652:

1,2,3,4,6,12,13,17,26,34,39,51,52,68,78,102,156,204,221,442,663,884,1326,2652


-1,-2,-3,-4,-6,-12,-13,-17,-26,-34,-39,-51,-52,-68,-78,-102,-156,-204,-221,-442,-663,-884,-1326,-2652 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -2652

(1)*(-2652)

(2)*(-1326)

(3)*(-884)

(4)*(-663)

(6)*(-442)

(12)*(-221)

(13)*(-204)

(17)*(-156)

(26)*(-102)

(34)*(-78)

(39)*(-68)

(51)*(-52)

(-1)*(2652)

(-2)*(1326)

(-3)*(884)

(-4)*(663)

(-6)*(442)

(-12)*(221)

(-13)*(204)

(-17)*(156)

(-26)*(102)

(-34)*(78)

(-39)*(68)

(-51)*(52)


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



Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-2652</td><td>1+(-2652)=-2651</td></tr><tr><td align="center">2</td><td align="center">-1326</td><td>2+(-1326)=-1324</td></tr><tr><td align="center">3</td><td align="center">-884</td><td>3+(-884)=-881</td></tr><tr><td align="center">4</td><td align="center">-663</td><td>4+(-663)=-659</td></tr><tr><td align="center">6</td><td align="center">-442</td><td>6+(-442)=-436</td></tr><tr><td align="center">12</td><td align="center">-221</td><td>12+(-221)=-209</td></tr><tr><td align="center">13</td><td align="center">-204</td><td>13+(-204)=-191</td></tr><tr><td align="center">17</td><td align="center">-156</td><td>17+(-156)=-139</td></tr><tr><td align="center">26</td><td align="center">-102</td><td>26+(-102)=-76</td></tr><tr><td align="center">34</td><td align="center">-78</td><td>34+(-78)=-44</td></tr><tr><td align="center">39</td><td align="center">-68</td><td>39+(-68)=-29</td></tr><tr><td align="center">51</td><td align="center">-52</td><td>51+(-52)=-1</td></tr><tr><td align="center">-1</td><td align="center">2652</td><td>-1+2652=2651</td></tr><tr><td align="center">-2</td><td align="center">1326</td><td>-2+1326=1324</td></tr><tr><td align="center">-3</td><td align="center">884</td><td>-3+884=881</td></tr><tr><td align="center">-4</td><td align="center">663</td><td>-4+663=659</td></tr><tr><td align="center">-6</td><td align="center">442</td><td>-6+442=436</td></tr><tr><td align="center">-12</td><td align="center">221</td><td>-12+221=209</td></tr><tr><td align="center">-13</td><td align="center">204</td><td>-13+204=191</td></tr><tr><td align="center">-17</td><td align="center">156</td><td>-17+156=139</td></tr><tr><td align="center">-26</td><td align="center">102</td><td>-26+102=76</td></tr><tr><td align="center">-34</td><td align="center">78</td><td>-34+78=44</td></tr><tr><td align="center">-39</td><td align="center">68</td><td>-39+68=29</td></tr><tr><td align="center">-51</td><td align="center">52</td><td>-51+52=1</td></tr></table>

None of these pairs of factors add to -5. So the expression {{{3x^2-5x-884}}} cannot be factored



In other words, the polynomial is prime.


    
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