Question 538206


{{{x^4-11x^3-12x^2}}} Start with the given expression.



{{{x^2(x^2-11x-12)}}} Factor out the GCF {{{x^2}}}.



Now let's try to factor the inner expression {{{x^2-11x-12}}}



---------------------------------------------------------------



Looking at the expression {{{x^2-11x-12}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-11}}}, and the last term is {{{-12}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-12}}} to get {{{(1)(-12)=-12}}}.



Now the question is: what two whole numbers multiply to {{{-12}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-11}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-12}}} (the previous product).



Factors of {{{-12}}}:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



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



These factors pair up and multiply to {{{-12}}}.

1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-11}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=red>1</font></td><td  align="center"><font color=red>-12</font></td><td  align="center"><font color=red>1+(-12)=-11</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>2+(-6)=-4</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>3+(-4)=-1</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-1+12=11</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-2+6=4</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-3+4=1</font></td></tr></table>



From the table, we can see that the two numbers {{{1}}} and {{{-12}}} add to {{{-11}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{-12}}} both multiply to {{{-12}}} <font size=4><b>and</b></font> add to {{{-11}}}



Now replace the middle term {{{-11x}}} with {{{x-12x}}}. Remember, {{{1}}} and {{{-12}}} add to {{{-11}}}. So this shows us that {{{x-12x=-11x}}}.



{{{x^2+highlight(x-12x)-12}}} Replace the second term {{{-11x}}} with {{{x-12x}}}.



{{{(x^2+x)+(-12x-12)}}} Group the terms into two pairs.



{{{x(x+1)+(-12x-12)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+1)-12(x+1)}}} Factor out {{{12}}} 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.



{{{(x-12)(x+1)}}} Combine like terms. Or factor out the common term {{{x+1}}}



--------------------------------------------------



So {{{x^2(x^2-11x-12)}}} then factors further to {{{x^2(x-12)(x+1)}}}



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



Answer:



So {{{x^4-11x^3-12x^2}}} completely factors to {{{x^2(x-12)(x+1)}}}.



In other words, {{{x^4-11x^3-12x^2=x^2(x-12)(x+1)}}}.



Note: you can check the answer by expanding {{{x^2(x-12)(x+1)}}} to get {{{x^4-11x^3-12x^2}}} or by graphing the original expression and the answer (the two graphs should be identical).



If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com">jim_thompson5910@hotmail.com</a>


Also, please consider visiting my website: <a href="http://www.freewebs.com/jimthompson5910/home.html">http://www.freewebs.com/jimthompson5910/home.html</a> and making a donation. Thank you


Jim