SOLUTION: factor completely 10x^2-29x-3

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Question 530219: factor completely 10x^2-29x-3
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression 10x%5E2-29x-3, we can see that the first coefficient is 10, the second coefficient is -29, and the last term is -3.


Now multiply the first coefficient 10 by the last term -3 to get %2810%29%28-3%29=-30.


Now the question is: what two whole numbers multiply to -30 (the previous product) and add to the second coefficient -29?


To find these two numbers, we need to list all of the factors of -30 (the previous product).


Factors of -30:
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


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


These factors pair up and multiply to -30.
1*(-30) = -30
2*(-15) = -30
3*(-10) = -30
5*(-6) = -30
(-1)*(30) = -30
(-2)*(15) = -30
(-3)*(10) = -30
(-5)*(6) = -30

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -29:


First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1



From the table, we can see that the two numbers 1 and -30 add to -29 (the middle coefficient).


So the two numbers 1 and -30 both multiply to -30 and add to -29


Now replace the middle term -29x with x-30x. Remember, 1 and -30 add to -29. So this shows us that x-30x=-29x.


10x%5E2%2Bhighlight%28x-30x%29-3 Replace the second term -29x with x-30x.


%2810x%5E2%2Bx%29%2B%28-30x-3%29 Group the terms into two pairs.


x%2810x%2B1%29%2B%28-30x-3%29 Factor out the GCF x from the first group.


x%2810x%2B1%29-3%2810x%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.


%28x-3%29%2810x%2B1%29 Combine like terms. Or factor out the common term 10x%2B1


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Answer:


So 10x%5E2-29x-3 factors to %28x-3%29%2810x%2B1%29.


In other words, 10x%5E2-29x-3=%28x-3%29%2810x%2B1%29.


Note: you can check the answer by expanding %28x-3%29%2810x%2B1%29 to get 10x%5E2-29x-3 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

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Jim