SOLUTION: in class we are factoring trinomials. 7a^3b-35a^2b^2+42ab^3

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: in class we are factoring trinomials. 7a^3b-35a^2b^2+42ab^3      Log On


   

Question 216205: in class we are factoring trinomials. 7a^3b-35a^2b^2+42ab^3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
7a%5E3b-35a%5E2b%5E2%2B42ab%5E3 Start with the given expression.


7ab%28a%5E2-5ab%2B6b%5E2%29 Factor out the GCF 7ab.


Now let's try to factor the inner expression a%5E2-5ab%2B6b%5E2


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


Looking at the expression a%5E2-5ab%2B6b%5E2, we can see that the first coefficient is 1, the second coefficient is -5, and the last coefficient is 6.


Now multiply the first coefficient 1 by the last coefficient 6 to get %281%29%286%29=6.


Now the question is: what two whole numbers multiply to 6 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of 6 (the previous product).


Factors of 6:
1,2,3,6
-1,-2,-3,-6


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


These factors pair up and multiply to 6.
1*6 = 6
2*3 = 6
(-1)*(-6) = 6
(-2)*(-3) = 6

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
161+6=7
232+3=5
-1-6-1+(-6)=-7
-2-3-2+(-3)=-5



From the table, we can see that the two numbers -2 and -3 add to -5 (the middle coefficient).


So the two numbers -2 and -3 both multiply to 6 and add to -5


Now replace the middle term -5ab with -2ab-3ab. Remember, -2 and -3 add to -5. So this shows us that -2ab-3ab=-5ab.


a%5E2%2Bhighlight%28-2ab-3ab%29%2B6b%5E2 Replace the second term -5ab with -2ab-3ab.


%28a%5E2-2ab%29%2B%28-3ab%2B6b%5E2%29 Group the terms into two pairs.


a%28a-2b%29%2B%28-3ab%2B6b%5E2%29 Factor out the GCF a from the first group.


a%28a-2b%29-3b%28a-2b%29 Factor out the GCF -3b from the second group.


%28a-3b%29%28a-2b%29 Combine like terms.


So this means that a%5E2-5ab%2B6b%5E2 factors to %28a-3b%29%28a-2b%29

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


So 7ab%28a%5E2-5ab%2B6b%5E2%29 factors to 7ab%28a-3b%29%28a-2b%29


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

Answer:


So 7a%5E3b-35a%5E2b%5E2%2B42ab%5E3 completely factors to 7ab%28a-3b%29%28a-2b%29