SOLUTION: THE factor polynomial problem is 15x2−84x−36 I know the answer to the polynomial is 3(5x+2)(x−6)but i dont understand how you get from 3(5x2−28x&#872

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: THE factor polynomial problem is 15x2−84x−36 I know the answer to the polynomial is 3(5x+2)(x−6)but i dont understand how you get from 3(5x2−28x&#872      Log On


   

Question 539493: THE factor polynomial problem is 15x2−84x−36
I know the answer to the polynomial is 3(5x+2)(x−6)but i dont understand how you get from
3(5x2−28x−12)to 3(5x+2)(x−6)<----the answer
can you please show me how you get it STEP BY STEP

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

15x%5E2-84x-36 Start with the given expression.


3%285x%5E2-28x-12%29 Factor out the GCF 3.


Now let's try to factor the inner expression 5x%5E2-28x-12


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Looking at the expression 5x%5E2-28x-12, we can see that the first coefficient is 5, the second coefficient is -28, and the last term is -12.


Now multiply the first coefficient 5 by the last term -12 to get %285%29%28-12%29=-60.


Now the question is: what two whole numbers multiply to -60 (the previous product) and add to the second coefficient -28?


To find these two numbers, we need to list all of the factors of -60 (the previous product).


Factors of -60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60


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


These factors pair up and multiply to -60.
1*(-60) = -60
2*(-30) = -60
3*(-20) = -60
4*(-15) = -60
5*(-12) = -60
6*(-10) = -60
(-1)*(60) = -60
(-2)*(30) = -60
(-3)*(20) = -60
(-4)*(15) = -60
(-5)*(12) = -60
(-6)*(10) = -60

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -28:


First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4



From the table, we can see that the two numbers 2 and -30 add to -28 (the middle coefficient).


So the two numbers 2 and -30 both multiply to -60 and add to -28


Now replace the middle term -28x with 2x-30x. Remember, 2 and -30 add to -28. So this shows us that 2x-30x=-28x.


5x%5E2%2Bhighlight%282x-30x%29-12 Replace the second term -28x with 2x-30x.


%285x%5E2%2B2x%29%2B%28-30x-12%29 Group the terms into two pairs.


x%285x%2B2%29%2B%28-30x-12%29 Factor out the GCF x from the first group.


x%285x%2B2%29-6%285x%2B2%29 Factor out 6 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-6%29%285x%2B2%29 Combine like terms. Or factor out the common term 5x%2B2


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So 3%285x%5E2-28x-12%29 then factors further to 3%28x-6%29%285x%2B2%29


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Answer:


So 15x%5E2-84x-36 completely factors to 3%28x-6%29%285x%2B2%29.


In other words, 15x%5E2-84x-36=3%28x-6%29%285x%2B2%29.


Note: you can check the answer by expanding 3%28x-6%29%285x%2B2%29 to get 15x%5E2-84x-36 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