SOLUTION: solve the polynomial 12x^2-5x-3

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Question 460669: solve the polynomial 12x^2-5x-3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you mean factor it.




Looking at the expression 12x%5E2-5x-3, we can see that the first coefficient is 12, the second coefficient is -5, and the last term is -3.


Now multiply the first coefficient 12 by the last term -3 to get %2812%29%28-3%29=-36.


Now the question is: what two whole numbers multiply to -36 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of -36 (the previous product).


Factors of -36:
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


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


These factors pair up and multiply to -36.
1*(-36) = -36
2*(-18) = -36
3*(-12) = -36
4*(-9) = -36
6*(-6) = -36
(-1)*(36) = -36
(-2)*(18) = -36
(-3)*(12) = -36
(-4)*(9) = -36
(-6)*(6) = -36

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


First NumberSecond NumberSum
1-361+(-36)=-35
2-182+(-18)=-16
3-123+(-12)=-9
4-94+(-9)=-5
6-66+(-6)=0
-136-1+36=35
-218-2+18=16
-312-3+12=9
-49-4+9=5
-66-6+6=0



From the table, we can see that the two numbers 4 and -9 add to -5 (the middle coefficient).


So the two numbers 4 and -9 both multiply to -36 and add to -5


Now replace the middle term -5x with 4x-9x. Remember, 4 and -9 add to -5. So this shows us that 4x-9x=-5x.


12x%5E2%2Bhighlight%284x-9x%29-3 Replace the second term -5x with 4x-9x.


%2812x%5E2%2B4x%29%2B%28-9x-3%29 Group the terms into two pairs.


4x%283x%2B1%29%2B%28-9x-3%29 Factor out the GCF 4x from the first group.


4x%283x%2B1%29-3%283x%2B1%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.


%284x-3%29%283x%2B1%29 Combine like terms. Or factor out the common term 3x%2B1


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Answer:


So 12x%5E2-5x-3 factors to %284x-3%29%283x%2B1%29.


In other words, 12x%5E2-5x-3=%284x-3%29%283x%2B1%29.


Note: you can check the answer by expanding %284x-3%29%283x%2B1%29 to get 12x%5E2-5x-3 or by graphing the original expression and the answer (the two graphs should be identical).