SOLUTION: Here is the question: 16(3x+2)^3+12(3x+2)^2-88(3x+2) Answer: 12x(3x+2)(12x+19) I cannot figure out how to get the 19!

Algebra ->  Expressions -> SOLUTION: Here is the question: 16(3x+2)^3+12(3x+2)^2-88(3x+2) Answer: 12x(3x+2)(12x+19) I cannot figure out how to get the 19!       Log On


   

Question 368253: Here is the question:
16(3x+2)^3+12(3x+2)^2-88(3x+2)
Answer: 12x(3x+2)(12x+19)
I cannot figure out how to get the 19!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
16%283x%2B2%29%5E3%2B12%283x%2B2%29%5E2-88%283x%2B2%29 Start with the given expression.


Let z=3x%2B2


16z%5E3%2B12z%5E2-88z Replace each 3x%2B2 term with 'z'



Looking at the expression 4z%5E2%2B3z-22, we can see that the first coefficient is 4, the second coefficient is 3, and the last term is -22.


Now multiply the first coefficient 4 by the last term -22 to get %284%29%28-22%29=-88.


Now the question is: what two whole numbers multiply to -88 (the previous product) and add to the second coefficient 3?


To find these two numbers, we need to list all of the factors of -88 (the previous product).


Factors of -88:
1,2,4,8,11,22,44,88
-1,-2,-4,-8,-11,-22,-44,-88


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


These factors pair up and multiply to -88.
1*(-88) = -88
2*(-44) = -88
4*(-22) = -88
8*(-11) = -88
(-1)*(88) = -88
(-2)*(44) = -88
(-4)*(22) = -88
(-8)*(11) = -88

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 3:


First NumberSecond NumberSum
1-881+(-88)=-87
2-442+(-44)=-42
4-224+(-22)=-18
8-118+(-11)=-3
-188-1+88=87
-244-2+44=42
-422-4+22=18
-811-8+11=3



From the table, we can see that the two numbers -8 and 11 add to 3 (the middle coefficient).


So the two numbers -8 and 11 both multiply to -88 and add to 3


Now replace the middle term 3z with -8z%2B11z. Remember, -8 and 11 add to 3. So this shows us that -8z%2B11z=3z.


4z%5E2%2Bhighlight%28-8z%2B11z%29-22 Replace the second term 3z with -8z%2B11z.


%284z%5E2-8z%29%2B%2811z-22%29 Group the terms into two pairs.


4z%28z-2%29%2B%2811z-22%29 Factor out the GCF 4z from the first group.


4z%28z-2%29%2B11%28z-2%29 Factor out 11 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.


%284z%2B11%29%28z-2%29 Combine like terms. Or factor out the common term z-2


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So 4z%284z%5E2%2B3z-22%29 then factors further to 4z%284z%2B11%29%28z-2%29



So 16z%5E3%2B12z%5E2-88z completely factors to 4z%284z%2B11%29%28z-2%29.


In other words, 16z%5E3%2B12z%5E2-88z=4z%284z%2B11%29%28z-2%29.


Now plug in z=3x%2B2 to get





Distribute.


16%283x%2B2%29%5E3%2B12%283x%2B2%29%5E2-88%283x%2B2%29=4%283x%2B2%29%2812x%2B19%29%283x%29 Combine like terms.


16%283x%2B2%29%5E3%2B12%283x%2B2%29%5E2-88%283x%2B2%29=4%283x%29%283x%2B2%29%2812x%2B19%29 Rearrange the terms.


16%283x%2B2%29%5E3%2B12%283x%2B2%29%5E2-88%283x%2B2%29=12x%283x%2B2%29%2812x%2B19%29 Multiply


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

Also, feel free to check out my tutoring website

Jim