SOLUTION: 2x^3+2x^2-4x over x^2-6x+9 simplify

Algebra ->  Equations -> SOLUTION: 2x^3+2x^2-4x over x^2-6x+9 simplify      Log On


   

Question 260263: 2x^3+2x^2-4x over x^2-6x+9 simplify
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
(2x(x^2+x-2))/(x-3)^2
2x%2A+%28x-1%29%2A%28x%2B2%29%2F%28x-3%29%5E2
Below are instructions on how to factor x^2+x-2 and x^2-6x+9
Well worth learning. They are long but easy to follow.
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


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



Now multiply the first coefficient 2 by the last term -2 to get %282%29%28-2%29=-4.



Now the question is: what two whole numbers multiply to -4 (the previous product) and add to the second coefficient 1?



To find these two numbers, we need to list all of the factors of -4 (the previous product).



Factors of -4:

1,2,4

-1,-2,-4



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



These factors pair up and multiply to -4.

1*(-4) = -4
2*(-2) = -4
(-1)*(4) = -4
(-2)*(2) = -4


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:



First NumberSecond NumberSum
1-41+(-4)=-3
2-22+(-2)=0
-14-1+4=3
-22-2+2=0




From the table, we can see that there are no pairs of numbers which add to 1. So 2x%5E2%2Bx-2 cannot be factored.



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





Answer:



So 2%2Ax%5E2%2Bx-2 doesn't factor at all (over the rational numbers).



So 2%2Ax%5E2%2Bx-2 is prime.



Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2-6x%2B9, we can see that the first coefficient is 1, the second coefficient is -6, and the last term is 9.



Now multiply the first coefficient 1 by the last term 9 to get %281%29%289%29=9.



Now the question is: what two whole numbers multiply to 9 (the previous product) and add to the second coefficient -6?



To find these two numbers, we need to list all of the factors of 9 (the previous product).



Factors of 9:

1,3,9

-1,-3,-9



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



These factors pair up and multiply to 9.

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


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



First NumberSecond NumberSum
191+9=10
333+3=6
-1-9-1+(-9)=-10
-3-3-3+(-3)=-6




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



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



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



x%5E2%2Bhighlight%28-3x-3x%29%2B9 Replace the second term -6x with -3x-3x.



%28x%5E2-3x%29%2B%28-3x%2B9%29 Group the terms into two pairs.



x%28x-3%29%2B%28-3x%2B9%29 Factor out the GCF x from the first group.



x%28x-3%29-3%28x-3%29 Factor out 3 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-3%29%28x-3%29 Combine like terms. Or factor out the common term x-3



%28x-3%29%5E2 Condense the terms.



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



Answer:



So x%5E2-6%2Ax%2B9 factors to %28x-3%29%5E2.



In other words, x%5E2-6%2Ax%2B9=%28x-3%29%5E2.



Note: you can check the answer by expanding %28x-3%29%5E2 to get x%5E2-6%2Ax%2B9 or by graphing the original expression and the answer (the two graphs should be identical).