SOLUTION: I could use some help with factoring this polynomial, 1/9x^2+2/3x-63 I hope someone can help Thank-You

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I could use some help with factoring this polynomial, 1/9x^2+2/3x-63 I hope someone can help Thank-You      Log On


   



Question 166139: I could use some help with factoring this polynomial,
1/9x^2+2/3x-63
I hope someone can help Thank-You

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2F9%29x%5E2%2B%282%2F3%29x-63 Start with the given expression


%281%2F9%29%28x%5E2%2B6x-567%29 Factor out the GCF 1%2F9. This will make every term on the inside of the parenthesis have a whole coefficient.


Now let's factor x%5E2%2B6x-567

---------------------------------------

Looking at the expression x%5E2%2B6x-567, we can see that the first coefficient is 1, the second coefficient is 6, and the last term is -567.


Now multiply the first coefficient 1 by the last term -567 to get %281%29%28-567%29=-567.


Now the question is: what two whole numbers multiply to -567 (the previous product) and add to the second coefficient 6?


To find these two numbers, we need to list all of the factors of -567 (the previous product).


Factors of -567:
1,3,7,9,21,27,63,81,189,567
-1,-3,-7,-9,-21,-27,-63,-81,-189,-567


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


These factors pair up and multiply to -567.
1*(-567)
3*(-189)
7*(-81)
9*(-63)
21*(-27)
(-1)*(567)
(-3)*(189)
(-7)*(81)
(-9)*(63)
(-21)*(27)

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


First NumberSecond NumberSum
1-5671+(-567)=-566
3-1893+(-189)=-186
7-817+(-81)=-74
9-639+(-63)=-54
21-2721+(-27)=-6
-1567-1+567=566
-3189-3+189=186
-781-7+81=74
-963-9+63=54
-2127-21+27=6



From the table, we can see that the two numbers -21 and 27 add to 6 (the middle coefficient).


So the two numbers -21 and 27 both multiply to -567 and add to 6


Now replace the middle term 6x with -21x%2B27x. Remember, -21 and 27 add to 6. So this shows us that -21x%2B27x=6x.


x%5E2%2Bhighlight%28-21x%2B27x%29-567 Replace the second term 6x with -21x%2B27x.


%28x%5E2-21x%29%2B%2827x-567%29 Group the terms into two pairs.


x%28x-21%29%2B%2827x-567%29 Factor out the GCF x from the first group.


x%28x-21%29%2B27%28x-21%29 Factor out 27 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%2B27%29%28x-21%29 Combine like terms. Or factor out the common term x-21





So x%5E2%2B6x-567 factors to %28x%2B27%29%28x-21%29.
---------------------------------------------


So %281%2F9%29%28x%5E2%2B6x-567%29 becomes %281%2F9%29%28x%2B27%29%28x-21%29



=============================================

Answer:

So %281%2F9%29x%5E2%2B%282%2F3%29x-63 completely factors to %281%2F9%29%28x%2B27%29%28x-21%29