Question 641517


Looking at the expression {{{36r^2-5r-24}}}, we can see that the first coefficient is {{{36}}}, the second coefficient is {{{-5}}}, and the last term is {{{-24}}}.



Now multiply the first coefficient {{{36}}} by the last term {{{-24}}} to get {{{(36)(-24)=-864}}}.



Now the question is: what two whole numbers multiply to {{{-864}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-5}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-864}}} (the previous product).



Factors of {{{-864}}}:

1,2,3,4,6,8,9,12,16,18,24,27,32,36,48,54,72,96,108,144,216,288,432,864

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-27,-32,-36,-48,-54,-72,-96,-108,-144,-216,-288,-432,-864



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-864}}}.

1*(-864) = -864
2*(-432) = -864
3*(-288) = -864
4*(-216) = -864
6*(-144) = -864
8*(-108) = -864
9*(-96) = -864
12*(-72) = -864
16*(-54) = -864
18*(-48) = -864
24*(-36) = -864
27*(-32) = -864
(-1)*(864) = -864
(-2)*(432) = -864
(-3)*(288) = -864
(-4)*(216) = -864
(-6)*(144) = -864
(-8)*(108) = -864
(-9)*(96) = -864
(-12)*(72) = -864
(-16)*(54) = -864
(-18)*(48) = -864
(-24)*(36) = -864
(-27)*(32) = -864


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-5}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-864</font></td><td  align="center"><font color=black>1+(-864)=-863</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-432</font></td><td  align="center"><font color=black>2+(-432)=-430</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-288</font></td><td  align="center"><font color=black>3+(-288)=-285</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-216</font></td><td  align="center"><font color=black>4+(-216)=-212</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-144</font></td><td  align="center"><font color=black>6+(-144)=-138</font></td></tr><tr><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>-108</font></td><td  align="center"><font color=black>8+(-108)=-100</font></td></tr><tr><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>-96</font></td><td  align="center"><font color=black>9+(-96)=-87</font></td></tr><tr><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-72</font></td><td  align="center"><font color=black>12+(-72)=-60</font></td></tr><tr><td  align="center"><font color=black>16</font></td><td  align="center"><font color=black>-54</font></td><td  align="center"><font color=black>16+(-54)=-38</font></td></tr><tr><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-48</font></td><td  align="center"><font color=black>18+(-48)=-30</font></td></tr><tr><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>-36</font></td><td  align="center"><font color=black>24+(-36)=-12</font></td></tr><tr><td  align="center"><font color=red>27</font></td><td  align="center"><font color=red>-32</font></td><td  align="center"><font color=red>27+(-32)=-5</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>864</font></td><td  align="center"><font color=black>-1+864=863</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>432</font></td><td  align="center"><font color=black>-2+432=430</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>288</font></td><td  align="center"><font color=black>-3+288=285</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>216</font></td><td  align="center"><font color=black>-4+216=212</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>144</font></td><td  align="center"><font color=black>-6+144=138</font></td></tr><tr><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>108</font></td><td  align="center"><font color=black>-8+108=100</font></td></tr><tr><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>96</font></td><td  align="center"><font color=black>-9+96=87</font></td></tr><tr><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>72</font></td><td  align="center"><font color=black>-12+72=60</font></td></tr><tr><td  align="center"><font color=black>-16</font></td><td  align="center"><font color=black>54</font></td><td  align="center"><font color=black>-16+54=38</font></td></tr><tr><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>48</font></td><td  align="center"><font color=black>-18+48=30</font></td></tr><tr><td  align="center"><font color=black>-24</font></td><td  align="center"><font color=black>36</font></td><td  align="center"><font color=black>-24+36=12</font></td></tr><tr><td  align="center"><font color=black>-27</font></td><td  align="center"><font color=black>32</font></td><td  align="center"><font color=black>-27+32=5</font></td></tr></table>



From the table, we can see that the two numbers {{{27}}} and {{{-32}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{27}}} and {{{-32}}} both multiply to {{{-864}}} <font size=4><b>and</b></font> add to {{{-5}}}



Now replace the middle term {{{-5r}}} with {{{27r-32r}}}. Remember, {{{27}}} and {{{-32}}} add to {{{-5}}}. So this shows us that {{{27r-32r=-5r}}}.



{{{36r^2+highlight(27r-32r)-24}}} Replace the second term {{{-5r}}} with {{{27r-32r}}}.



{{{(36r^2+27r)+(-32r-24)}}} Group the terms into two pairs.



{{{9r(4r+3)+(-32r-24)}}} Factor out the GCF {{{9r}}} from the first group.



{{{9r(4r+3)-8(4r+3)}}} Factor out {{{8}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(9r-8)(4r+3)}}} Combine like terms. Or factor out the common term {{{4r+3}}}



===============================================================



Answer:



So {{{36r^2-5r-24}}} factors to {{{(9r-8)(4r+3)}}}.



In other words, {{{36r^2-5r-24=(9r-8)(4r+3)}}}.



Note: you can check the answer by expanding {{{(9r-8)(4r+3)}}} to get {{{36r^2-5r-24}}} or by graphing the original expression and the answer (the two graphs should be identical).


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