SOLUTION: I'm having a problem in figuring out what the "2" in front of 2x(squared) means. This is factoring polynomials mentally or with the box method. 2x(squared) + 11x - 21 The mi

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I'm having a problem in figuring out what the "2" in front of 2x(squared) means. This is factoring polynomials mentally or with the box method. 2x(squared) + 11x - 21 The mi      Log On


   

Question 725344: I'm having a problem in figuring out what the "2" in front of 2x(squared) means.
This is factoring polynomials mentally or with the box method.
2x(squared) + 11x - 21
The missing numbers are -3 and 14.
I know I'm supposed to multiply -21 by 2, but what do I do after solving for the two missing numbers?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 2x%5E2%2B11x-21, we can see that the first coefficient is 2, the second coefficient is 11, and the last term is -21.


Now multiply the first coefficient 2 by the last term -21 to get %282%29%28-21%29=-42.


Now the question is: what two whole numbers multiply to -42 (the previous product) and add to the second coefficient 11?


To find these two numbers, we need to list all of the factors of -42 (the previous product).


Factors of -42:
1,2,3,6,7,14,21,42
-1,-2,-3,-6,-7,-14,-21,-42


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


These factors pair up and multiply to -42.
1*(-42) = -42
2*(-21) = -42
3*(-14) = -42
6*(-7) = -42
(-1)*(42) = -42
(-2)*(21) = -42
(-3)*(14) = -42
(-6)*(7) = -42

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


First NumberSecond NumberSum
1-421+(-42)=-41
2-212+(-21)=-19
3-143+(-14)=-11
6-76+(-7)=-1
-142-1+42=41
-221-2+21=19
-314-3+14=11
-67-6+7=1



From the table, we can see that the two numbers -3 and 14 add to 11 (the middle coefficient).


So the two numbers -3 and 14 both multiply to -42 and add to 11


Now replace the middle term 11x with -3x%2B14x. Remember, -3 and 14 add to 11. So this shows us that -3x%2B14x=11x.


2x%5E2%2Bhighlight%28-3x%2B14x%29-21 Replace the second term 11x with -3x%2B14x.


%282x%5E2-3x%29%2B%2814x-21%29 Group the terms into two pairs.


x%282x-3%29%2B%2814x-21%29 Factor out the GCF x from the first group.


x%282x-3%29%2B7%282x-3%29 Factor out 7 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%2B7%29%282x-3%29 Combine like terms. Or factor out the common term 2x-3


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Answer:


So 2x%5E2%2B11x-21 factors to %28x%2B7%29%282x-3%29.


In other words, 2x%5E2%2B11x-21=%28x%2B7%29%282x-3%29.


Note: you can check the answer by expanding %28x%2B7%29%282x-3%29 to get 2x%5E2%2B11x-21 or by graphing the original expression and the answer (the two graphs should be identical).