SOLUTION: Factor the following polynomial completely 14x^2 - 20x + 6 Can you explain the process to me on how to figure this out. I am a little lost.

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Question 101290: Factor the following polynomial completely
14x^2 - 20x + 6
Can you explain the process to me on how to figure this out. I am a little lost.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


14%2Ax%5E2-20%2Ax%2B6 Start with the given expression.



2%287x%5E2-10x%2B3%29 Factor out the GCF 2.



Now let's try to factor the inner expression 7x%5E2-10x%2B3



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Looking at the expression 7x%5E2-10x%2B3, we can see that the first coefficient is 7, the second coefficient is -10, and the last term is 3.



Now multiply the first coefficient 7 by the last term 3 to get %287%29%283%29=21.



Now the question is: what two whole numbers multiply to 21 (the previous product) and add to the second coefficient -10?



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 -10:



First NumberSecond NumberSum
1211+21=22
373+7=10
-1-21-1+(-21)=-22
-3-7-3+(-7)=-10




From the table, we can see that the two numbers -3 and -7 add to -10 (the middle coefficient).



So the two numbers -3 and -7 both multiply to 21 and add to -10



Now replace the middle term -10x with -3x-7x. Remember, -3 and -7 add to -10. So this shows us that -3x-7x=-10x.



7x%5E2%2Bhighlight%28-3x-7x%29%2B3 Replace the second term -10x with -3x-7x.



%287x%5E2-3x%29%2B%28-7x%2B3%29 Group the terms into two pairs.



x%287x-3%29%2B%28-7x%2B3%29 Factor out the GCF x from the first group.



x%287x-3%29-1%287x-3%29 Factor out 1 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-1%29%287x-3%29 Combine like terms. Or factor out the common term 7x-3



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So 2%287x%5E2-10x%2B3%29 then factors further to 2%28x-1%29%287x-3%29



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Answer:



So 14%2Ax%5E2-20%2Ax%2B6 completely factors to 2%28x-1%29%287x-3%29.



In other words, 14%2Ax%5E2-20%2Ax%2B6=2%28x-1%29%287x-3%29.



Note: you can check the answer by expanding 2%28x-1%29%287x-3%29 to get 14%2Ax%5E2-20%2Ax%2B6 or by graphing the original expression and the answer (the two graphs should be identical).