Question 645131
Note:


6x^3 + 15x^2 + -9x


is the same as 


6x^3 + 15x^2 - 9x



{{{6x^3+15x^2-9x}}} Start with the given expression.



{{{3x(2x^2+5x-3)}}} Factor out the GCF {{{3x}}}.



Now let's try to factor the inner expression {{{2x^2+5x-3}}}



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Looking at the expression {{{2x^2+5x-3}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{5}}}, and the last term is {{{-3}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{-3}}} to get {{{(2)(-3)=-6}}}.



Now the question is: what two whole numbers multiply to {{{-6}}} (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 {{{-6}}} (the previous product).



Factors of {{{-6}}}:

1,2,3,6

-1,-2,-3,-6



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



These factors pair up and multiply to {{{-6}}}.

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


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>-6</font></td><td  align="center"><font color=black>1+(-6)=-5</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>2+(-3)=-1</font></td></tr><tr><td  align="center"><font color=red>-1</font></td><td  align="center"><font color=red>6</font></td><td  align="center"><font color=red>-1+6=5</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-2+3=1</font></td></tr></table>



From the table, we can see that the two numbers {{{-1}}} and {{{6}}} add to {{{5}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{6}}} both multiply to {{{-6}}} <font size=4><b>and</b></font> add to {{{5}}}



Now replace the middle term {{{5x}}} with {{{-x+6x}}}. Remember, {{{-1}}} and {{{6}}} add to {{{5}}}. So this shows us that {{{-x+6x=5x}}}.



{{{2x^2+highlight(-x+6x)-3}}} Replace the second term {{{5x}}} with {{{-x+6x}}}.



{{{(2x^2-x)+(6x-3)}}} Group the terms into two pairs.



{{{x(2x-1)+(6x-3)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(2x-1)+3(2x-1)}}} Factor out {{{3}}} 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+3)(2x-1)}}} Combine like terms. Or factor out the common term {{{2x-1}}}



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So {{{3x(2x^2+5x-3)}}} then factors further to {{{3x(x+3)(2x-1)}}}



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Answer:



So {{{6x^3+15x^2-9x}}} completely factors to {{{3x(x+3)(2x-1)}}}.



In other words, {{{6x^3+15x^2-9x=3x(x+3)(2x-1)}}} for all values of x.



Note: you can check the answer by expanding {{{3x(x+3)(2x-1)}}} to get {{{6x^3+15x^2-9x}}} or by graphing the original expression and the answer (the two graphs should be identical).


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