Question 303320
I'll do the first two to get you started. Please only post one problem at a time.



a)


{{{v^3-5v^2-36v}}} Start with the given expression.



{{{v(v^2-5v-36)}}} Factor out the GCF {{{v}}}.



Now let's try to factor the inner expression {{{v^2-5v-36}}}



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Looking at the expression {{{v^2-5v-36}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-5}}}, and the last term is {{{-36}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-36}}} to get {{{(1)(-36)=-36}}}.



Now the question is: what two whole numbers multiply to {{{-36}}} (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 {{{-36}}} (the previous product).



Factors of {{{-36}}}:

1,2,3,4,6,9,12,18,36

-1,-2,-3,-4,-6,-9,-12,-18,-36



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



These factors pair up and multiply to {{{-36}}}.

1*(-36) = -36
2*(-18) = -36
3*(-12) = -36
4*(-9) = -36
6*(-6) = -36
(-1)*(36) = -36
(-2)*(18) = -36
(-3)*(12) = -36
(-4)*(9) = -36
(-6)*(6) = -36


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>-36</font></td><td  align="center"><font color=black>1+(-36)=-35</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>2+(-18)=-16</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>3+(-12)=-9</font></td></tr><tr><td  align="center"><font color=red>4</font></td><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>4+(-9)=-5</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>6+(-6)=0</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>36</font></td><td  align="center"><font color=black>-1+36=35</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>-2+18=16</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-3+12=9</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>-4+9=5</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-6+6=0</font></td></tr></table>



From the table, we can see that the two numbers {{{4}}} and {{{-9}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{4}}} and {{{-9}}} both multiply to {{{-36}}} <font size=4><b>and</b></font> add to {{{-5}}}



Now replace the middle term {{{-5v}}} with {{{4v-9v}}}. Remember, {{{4}}} and {{{-9}}} add to {{{-5}}}. So this shows us that {{{4v-9v=-5v}}}.



{{{v^2+highlight(4v-9v)-36}}} Replace the second term {{{-5v}}} with {{{4v-9v}}}.



{{{(v^2+4v)+(-9v-36)}}} Group the terms into two pairs.



{{{v(v+4)+(-9v-36)}}} Factor out the GCF {{{v}}} from the first group.



{{{v(v+4)-9(v+4)}}} Factor out {{{9}}} 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.



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



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So {{{v(v^2-5v-36)}}} then factors further to {{{v(v-9)(v+4)}}}



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Answer:



So {{{v^3-5v^2-36v}}} completely factors to {{{v(v-9)(v+4)}}}.



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



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



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b)


{{{r^2+3r-18=0}}} Start with the given equation


{{{(r+6)(r-3)=0}}} <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>Factor</a> the left side 




Now set each factor equal to zero:

{{{r+6=0}}} or  {{{r-3=0}}} 


{{{r=-6}}} or  {{{r=3}}}    Now solve for r in each case



So the solutions are {{{r=-6}}} or  {{{r=3}}}