SOLUTION: Use the ac test to determine which of the following trinomials can be factored. Find the values of m and n for each trinomial that can be factored. x squared + 2x - 15

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Use the ac test to determine which of the following trinomials can be factored. Find the values of m and n for each trinomial that can be factored. x squared + 2x - 15      Log On


   

Question 114757: Use the ac test to determine which of the following trinomials can be factored. Find the values of m and n for each trinomial that can be factored.
x squared + 2x - 15
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)


Looking at the expression x%5E2%2B2x-15, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is -15.



Now multiply the first coefficient 1 by the last term -15 to get %281%29%28-15%29=-15.



Now the question is: what two whole numbers multiply to -15 (the previous product) and add to the second coefficient 2?



To find these two numbers, we need to list all of the factors of -15 (the previous product).



Factors of -15:

1,3,5,15

-1,-3,-5,-15



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



These factors pair up and multiply to -15.

1*(-15) = -15
3*(-5) = -15
(-1)*(15) = -15
(-3)*(5) = -15


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:



First NumberSecond NumberSum
1-151+(-15)=-14
3-53+(-5)=-2
-115-1+15=14
-35-3+5=2




From the table, we can see that the two numbers -3 and 5 add to 2 (the middle coefficient).



So the two numbers -3 and 5 both multiply to -15 and add to 2



Now replace the middle term 2x with -3x%2B5x. Remember, -3 and 5 add to 2. So this shows us that -3x%2B5x=2x.



x%5E2%2Bhighlight%28-3x%2B5x%29-15 Replace the second term 2x with -3x%2B5x.



%28x%5E2-3x%29%2B%285x-15%29 Group the terms into two pairs.



x%28x-3%29%2B%285x-15%29 Factor out the GCF x from the first group.



x%28x-3%29%2B5%28x-3%29 Factor out 5 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%2B5%29%28x-3%29 Combine like terms. Or factor out the common term x-3



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Answer:



So x%5E2%2B2%2Ax-15 factors to %28x%2B5%29%28x-3%29.



In other words, x%5E2%2B2%2Ax-15=%28x%2B5%29%28x-3%29.



Note: you can check the answer by expanding %28x%2B5%29%28x-3%29 to get x%5E2%2B2%2Ax-15 or by graphing the original expression and the answer (the two graphs should be identical).