SOLUTION: How do I completely factor the polynomial x^2-6x-27

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Question 112183: How do I completely factor the polynomial x^2-6x-27
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2-6x-27, we can see that the first coefficient is 1, the second coefficient is -6, and the last term is -27.



Now multiply the first coefficient 1 by the last term -27 to get %281%29%28-27%29=-27.



Now the question is: what two whole numbers multiply to -27 (the previous product) and add to the second coefficient -6?



To find these two numbers, we need to list all of the factors of -27 (the previous product).



Factors of -27:

1,3,9,27

-1,-3,-9,-27



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



These factors pair up and multiply to -27.

1*(-27) = -27
3*(-9) = -27
(-1)*(27) = -27
(-3)*(9) = -27


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -6:



First NumberSecond NumberSum
1-271+(-27)=-26
3-93+(-9)=-6
-127-1+27=26
-39-3+9=6




From the table, we can see that the two numbers 3 and -9 add to -6 (the middle coefficient).



So the two numbers 3 and -9 both multiply to -27 and add to -6



Now replace the middle term -6x with 3x-9x. Remember, 3 and -9 add to -6. So this shows us that 3x-9x=-6x.



x%5E2%2Bhighlight%283x-9x%29-27 Replace the second term -6x with 3x-9x.



%28x%5E2%2B3x%29%2B%28-9x-27%29 Group the terms into two pairs.



x%28x%2B3%29%2B%28-9x-27%29 Factor out the GCF x from the first group.



x%28x%2B3%29-9%28x%2B3%29 Factor out 9 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.



%28x-9%29%28x%2B3%29 Combine like terms. Or factor out the common term x%2B3



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Answer:



So x%5E2-6%2Ax-27 factors to %28x-9%29%28x%2B3%29.



In other words, x%5E2-6%2Ax-27=%28x-9%29%28x%2B3%29.



Note: you can check the answer by expanding %28x-9%29%28x%2B3%29 to get x%5E2-6%2Ax-27 or by graphing the original expression and the answer (the two graphs should be identical).