SOLUTION: Factor completely. If a polynomial is prime, state this. -12x^2-28x+24

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Question 610228: Factor completely. If a polynomial is prime, state this.
-12x^2-28x+24
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

-12x%5E2-28x%2B24 Start with the given expression.


-4%283x%5E2%2B7x-6%29 Factor out the GCF -4.


Now let's try to factor the inner expression 3x%5E2%2B7x-6


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Looking at the expression 3x%5E2%2B7x-6, we can see that the first coefficient is 3, the second coefficient is 7, and the last term is -6.


Now multiply the first coefficient 3 by the last term -6 to get %283%29%28-6%29=-18.


Now the question is: what two whole numbers multiply to -18 (the previous product) and add to the second coefficient 7?


To find these two numbers, we need to list all of the factors of -18 (the previous product).


Factors of -18:
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:


First NumberSecond NumberSum
1-181+(-18)=-17
2-92+(-9)=-7
3-63+(-6)=-3
-118-1+18=17
-29-2+9=7
-36-3+6=3



From the table, we can see that the two numbers -2 and 9 add to 7 (the middle coefficient).


So the two numbers -2 and 9 both multiply to -18 and add to 7


Now replace the middle term 7x with -2x%2B9x. Remember, -2 and 9 add to 7. So this shows us that -2x%2B9x=7x.


3x%5E2%2Bhighlight%28-2x%2B9x%29-6 Replace the second term 7x with -2x%2B9x.


%283x%5E2-2x%29%2B%289x-6%29 Group the terms into two pairs.


x%283x-2%29%2B%289x-6%29 Factor out the GCF x from the first group.


x%283x-2%29%2B3%283x-2%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%2B3%29%283x-2%29 Combine like terms. Or factor out the common term 3x-2


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So -4%283x%5E2%2B7x-6%29 then factors further to -4%28x%2B3%29%283x-2%29


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Answer:


So -12x%5E2-28x%2B24 completely factors to -4%28x%2B3%29%283x-2%29.


In other words, -12x%5E2-28x%2B24=-4%28x%2B3%29%283x-2%29.


Note: you can check the answer by expanding -4%28x%2B3%29%283x-2%29 to get -12x%5E2-28x%2B24 or by graphing the original expression and the answer (the two graphs should be identical).

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