SOLUTION: 3x^2+12x-63= How do i factor this polynomial?

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Question 777814: 3x^2+12x-63=
How do i factor this polynomial?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

3x%5E2%2B12x-63 Start with the given expression.


3%28x%5E2%2B4x-21%29 Factor out the GCF 3.


Now let's try to factor the inner expression x%5E2%2B4x-21


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Looking at the expression x%5E2%2B4x-21, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is -21.


Now multiply the first coefficient 1 by the last term -21 to get %281%29%28-21%29=-21.


Now the question is: what two whole numbers multiply to -21 (the previous product) and add to the second coefficient 4?


To find these two numbers, we need to list all of the factors of -21 (the previous product).


Factors of -21:
1,3,7,21
-1,-3,-7,-21


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:


First NumberSecond NumberSum
1-211+(-21)=-20
3-73+(-7)=-4
-121-1+21=20
-37-3+7=4



From the table, we can see that the two numbers -3 and 7 add to 4 (the middle coefficient).


So the two numbers -3 and 7 both multiply to -21 and add to 4


Now replace the middle term 4x with -3x%2B7x. Remember, -3 and 7 add to 4. So this shows us that -3x%2B7x=4x.


x%5E2%2Bhighlight%28-3x%2B7x%29-21 Replace the second term 4x with -3x%2B7x.


%28x%5E2-3x%29%2B%287x-21%29 Group the terms into two pairs.


x%28x-3%29%2B%287x-21%29 Factor out the GCF x from the first group.


x%28x-3%29%2B7%28x-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%28x-3%29 Combine like terms. Or factor out the common term x-3


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So 3%28x%5E2%2B4x-21%29 then factors further to 3%28x%2B7%29%28x-3%29


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Answer:


So 3x%5E2%2B12x-63 completely factors to 3%28x%2B7%29%28x-3%29.


In other words, 3x%5E2%2B12x-63=3%28x%2B7%29%28x-3%29.


Note: you can check the answer by expanding 3%28x%2B7%29%28x-3%29 to get 3x%5E2%2B12x-63 or by graphing the original expression and the answer (the two graphs should be identical).