SOLUTION: Factor the polynomial completely. If the polynomial cannot be factored, choose prime 7x^2 - 9x - 10

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Question 102391: Factor the polynomial completely. If the polynomial cannot be factored, choose prime
7x^2 - 9x - 10
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 7x%5E2-9x-10, we can see that the first coefficient is 7, the second coefficient is -9, and the last term is -10.



Now multiply the first coefficient 7 by the last term -10 to get %287%29%28-10%29=-70.



Now the question is: what two whole numbers multiply to -70 (the previous product) and add to the second coefficient -9?



To find these two numbers, we need to list all of the factors of -70 (the previous product).



Factors of -70:

1,2,5,7,10,14,35,70

-1,-2,-5,-7,-10,-14,-35,-70



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



These factors pair up and multiply to -70.

1*(-70) = -70
2*(-35) = -70
5*(-14) = -70
7*(-10) = -70
(-1)*(70) = -70
(-2)*(35) = -70
(-5)*(14) = -70
(-7)*(10) = -70


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -9:



First NumberSecond NumberSum
1-701+(-70)=-69
2-352+(-35)=-33
5-145+(-14)=-9
7-107+(-10)=-3
-170-1+70=69
-235-2+35=33
-514-5+14=9
-710-7+10=3




From the table, we can see that the two numbers 5 and -14 add to -9 (the middle coefficient).



So the two numbers 5 and -14 both multiply to -70 and add to -9



Now replace the middle term -9x with 5x-14x. Remember, 5 and -14 add to -9. So this shows us that 5x-14x=-9x.



7x%5E2%2Bhighlight%285x-14x%29-10 Replace the second term -9x with 5x-14x.



%287x%5E2%2B5x%29%2B%28-14x-10%29 Group the terms into two pairs.



x%287x%2B5%29%2B%28-14x-10%29 Factor out the GCF x from the first group.



x%287x%2B5%29-2%287x%2B5%29 Factor out 2 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-2%29%287x%2B5%29 Combine like terms. Or factor out the common term 7x%2B5



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Answer:



So 7%2Ax%5E2-9%2Ax-10 factors to %28x-2%29%287x%2B5%29.



In other words, 7%2Ax%5E2-9%2Ax-10=%28x-2%29%287x%2B5%29.



Note: you can check the answer by expanding %28x-2%29%287x%2B5%29 to get 7%2Ax%5E2-9%2Ax-10 or by graphing the original expression and the answer (the two graphs should be identical).