Question 578009


{{{9x^7-45x^6+18x^5}}} Start with the given expression.



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



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



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



Now multiply the first coefficient {{{1}}} by the last term {{{2}}} to get {{{(1)(2)=2}}}.



Now the question is: what two whole numbers multiply to {{{2}}} (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 {{{2}}} (the previous product).



Factors of {{{2}}}:

1,2

-1,-2



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



These factors pair up and multiply to {{{2}}}.

1*2 = 2
(-1)*(-2) = 2


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>2</font></td><td  align="center"><font color=black>1+2=3</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-1+(-2)=-3</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+2}}} cannot be factored.



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Answer:



So {{{9x^7-45x^6+18x^5}}} simply factors to {{{9x^5(x^2-5x+2)}}}



In other words, {{{9x^7-45x^6+18x^5=9x^5(x^2-5x+2)}}}.