SOLUTION: Factoring Completrly: x^3+7x^2-x-7 The ac method: 3x^2+11x+10

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Question 884723: Factoring Completrly: x^3+7x^2-x-7
The ac method: 3x^2+11x+10
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Problem #1: Factoring x%5E3%2B7x%5E2-x-7


x%5E3%2B7x%5E2-x-7 Start with the given expression.


%28x%5E3%2B7x%5E2%29%2B%28-x-7%29 Group the terms


x%5E2%28x%2B7%29%2B%28-x-7%29 Factor out x%5E2 from the first group


x%5E2%28x%2B7%29-1%28x%2B7%29 Factor out -1 from the second group


%28x%5E2-1%29%28x%2B7%29 Factor out x%2B7


%28x-1%29%28x%2B1%29%28x%2B7%29 Factor x%5E2-1 to get %28x-1%29%28x%2B1%29 (difference of squares rule)


So x%5E3%2B7x%5E2-x-7 factors to %28x-1%29%28x%2B1%29%28x%2B7%29

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Problem # 2: Factoring 3x%5E2%2B11x%2B10


Looking at the expression 3x%5E2%2B11x%2B10, we can see that the first coefficient is 3, the second coefficient is 11, and the last term is 10.


Now multiply the first coefficient 3 by the last term 10 to get %283%29%2810%29=30.


Now the question is: what two whole numbers multiply to 30 (the previous product) and add to the second coefficient 11?


To find these two numbers, we need to list all of the factors of 30 (the previous product).


Factors of 30:
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


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


These factors pair up and multiply to 30.
1*30 = 30
2*15 = 30
3*10 = 30
5*6 = 30
(-1)*(-30) = 30
(-2)*(-15) = 30
(-3)*(-10) = 30
(-5)*(-6) = 30

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 11:


First NumberSecond NumberSum
1301+30=31
2152+15=17
3103+10=13
565+6=11
-1-30-1+(-30)=-31
-2-15-2+(-15)=-17
-3-10-3+(-10)=-13
-5-6-5+(-6)=-11



From the table, we can see that the two numbers 5 and 6 add to 11 (the middle coefficient).


So the two numbers 5 and 6 both multiply to 30 and add to 11


Now replace the middle term 11x with 5x%2B6x. Remember, 5 and 6 add to 11. So this shows us that 5x%2B6x=11x.


3x%5E2%2Bhighlight%285x%2B6x%29%2B10 Replace the second term 11x with 5x%2B6x.


%283x%5E2%2B5x%29%2B%286x%2B10%29 Group the terms into two pairs.


x%283x%2B5%29%2B%286x%2B10%29 Factor out the GCF x from the first group.


x%283x%2B5%29%2B2%283x%2B5%29 Factor out 2 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%2B2%29%283x%2B5%29 Combine like terms. Or factor out the common term 3x%2B5


So 3x%5E2%2B11x%2B10 factors to %28x%2B2%29%283x%2B5%29.


In other words, 3x%5E2%2B11x%2B10=%28x%2B2%29%283x%2B5%29.


Note: you can check the answer by expanding %28x%2B2%29%283x%2B5%29 to get 3x%5E2%2B11x%2B10 or by graphing the original expression and the answer (the two graphs should be identical).