Question 635633


{{{9x^5-45x^4+27x^3}}} Start with the given expression.



{{{9x^3(x^2-5x+3)}}} Factor out the GCF {{{9x^3}}}.



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



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Looking at the expression {{{x^2-5x+3}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-5}}}, and the last term is {{{3}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{3}}} to get {{{(1)(3)=3}}}.



Now the question is: what two whole numbers multiply to {{{3}}} (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 {{{3}}} (the previous product).



Factors of {{{3}}}:

1,3

-1,-3



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



These factors pair up and multiply to {{{3}}}.

1*3 = 3
(-1)*(-3) = 3


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>3</font></td><td  align="center"><font color=black>1+3=4</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-1+(-3)=-4</font></td></tr></table>



From the table, we can see that there are no pairs of numbers which add to {{{-5}}}. So {{{x^2-5x+3}}} cannot be factored.



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Answer:



So {{{9x^5-45x^4+27x^3}}} simply factors to {{{9x^3(x^2-5x+3)}}}



In other words, {{{9x^5-45x^4+27x^3=9x^3(x^2-5x+3)}}}.