Question 315391


{{{36x^4-8x^3-28x^2}}} Start with the given expression.



{{{4x^2(9x^2-2x-7)}}} Factor out the GCF {{{4x^2}}}.



Now let's try to factor the inner expression {{{9x^2-2x-7}}}



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Looking at the expression {{{9x^2-2x-7}}}, we can see that the first coefficient is {{{9}}}, the second coefficient is {{{-2}}}, and the last term is {{{-7}}}.



Now multiply the first coefficient {{{9}}} by the last term {{{-7}}} to get {{{(9)(-7)=-63}}}.



Now the question is: what two whole numbers multiply to {{{-63}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-2}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-63}}} (the previous product).



Factors of {{{-63}}}:

1,3,7,9,21,63

-1,-3,-7,-9,-21,-63



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



These factors pair up and multiply to {{{-63}}}.

1*(-63) = -63
3*(-21) = -63
7*(-9) = -63
(-1)*(63) = -63
(-3)*(21) = -63
(-7)*(9) = -63


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-2}}}:



<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>-63</font></td><td  align="center"><font color=black>1+(-63)=-62</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-21</font></td><td  align="center"><font color=black>3+(-21)=-18</font></td></tr><tr><td  align="center"><font color=red>7</font></td><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>7+(-9)=-2</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>63</font></td><td  align="center"><font color=black>-1+63=62</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>21</font></td><td  align="center"><font color=black>-3+21=18</font></td></tr><tr><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>-7+9=2</font></td></tr></table>



From the table, we can see that the two numbers {{{7}}} and {{{-9}}} add to {{{-2}}} (the middle coefficient).



So the two numbers {{{7}}} and {{{-9}}} both multiply to {{{-63}}} <font size=4><b>and</b></font> add to {{{-2}}}



Now replace the middle term {{{-2x}}} with {{{7x-9x}}}. Remember, {{{7}}} and {{{-9}}} add to {{{-2}}}. So this shows us that {{{7x-9x=-2x}}}.



{{{9x^2+highlight(7x-9x)-7}}} Replace the second term {{{-2x}}} with {{{7x-9x}}}.



{{{(9x^2+7x)+(-9x-7)}}} Group the terms into two pairs.



{{{x(9x+7)+(-9x-7)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(9x+7)-1(9x+7)}}} Factor out {{{1}}} 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-1)(9x+7)}}} Combine like terms. Or factor out the common term {{{9x+7}}}



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So {{{4x^2(9x^2-2x-7)}}} then factors further to {{{4x^2(x-1)(9x+7)}}}



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Answer:



So {{{36x^4-8x^3-28x^2}}} completely factors to {{{4x^2(x-1)(9x+7)}}}.



In other words, {{{36x^4-8x^3-28x^2=4x^2(x-1)(9x+7)}}}.



Note: you can check the answer by expanding {{{4x^2(x-1)(9x+7)}}} to get {{{36x^4-8x^3-28x^2}}} or by graphing the original expression and the answer (the two graphs should be identical).