SOLUTION: Which of the following is one of the two binomial factors of 6s2 + 40s – 64?

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Which of the following is one of the two binomial factors of 6s2 + 40s – 64?      Log On


   

Question 522895: Which of the following is one of the two binomial factors of 6s2 + 40s – 64?
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

6s%5E2%2B40s-64 Start with the given expression.


2%283s%5E2%2B20s-32%29 Factor out the GCF 2.


Now let's try to factor the inner expression 3s%5E2%2B20s-32


---------------------------------------------------------------


Looking at the expression 3s%5E2%2B20s-32, we can see that the first coefficient is 3, the second coefficient is 20, and the last term is -32.


Now multiply the first coefficient 3 by the last term -32 to get %283%29%28-32%29=-96.


Now the question is: what two whole numbers multiply to -96 (the previous product) and add to the second coefficient 20?


To find these two numbers, we need to list all of the factors of -96 (the previous product).


Factors of -96:
1,2,3,4,6,8,12,16,24,32,48,96
-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96


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


These factors pair up and multiply to -96.
1*(-96) = -96
2*(-48) = -96
3*(-32) = -96
4*(-24) = -96
6*(-16) = -96
8*(-12) = -96
(-1)*(96) = -96
(-2)*(48) = -96
(-3)*(32) = -96
(-4)*(24) = -96
(-6)*(16) = -96
(-8)*(12) = -96

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 20:


First NumberSecond NumberSum
1-961+(-96)=-95
2-482+(-48)=-46
3-323+(-32)=-29
4-244+(-24)=-20
6-166+(-16)=-10
8-128+(-12)=-4
-196-1+96=95
-248-2+48=46
-332-3+32=29
-424-4+24=20
-616-6+16=10
-812-8+12=4



From the table, we can see that the two numbers -4 and 24 add to 20 (the middle coefficient).


So the two numbers -4 and 24 both multiply to -96 and add to 20


Now replace the middle term 20s with -4s%2B24s. Remember, -4 and 24 add to 20. So this shows us that -4s%2B24s=20s.


3s%5E2%2Bhighlight%28-4s%2B24s%29-32 Replace the second term 20s with -4s%2B24s.


%283s%5E2-4s%29%2B%2824s-32%29 Group the terms into two pairs.


s%283s-4%29%2B%2824s-32%29 Factor out the GCF s from the first group.


s%283s-4%29%2B8%283s-4%29 Factor out 8 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.


%28s%2B8%29%283s-4%29 Combine like terms. Or factor out the common term 3s-4


--------------------------------------------------


So 2%283s%5E2%2B20s-32%29 then factors further to 2%28s%2B8%29%283s-4%29


===============================================================


Answer:


So 6s%5E2%2B40s-64 completely factors to 2%28s%2B8%29%283s-4%29.


In other words, 6s%5E2%2B40s-64=2%28s%2B8%29%283s-4%29.


Note: you can check the answer by expanding 2%28s%2B8%29%283s-4%29 to get 6s%5E2%2B40s-64 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

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Which of the following is one of the
two binomial factors of 6s^2 + 40s – 64?
----
Common Factor: 2
2[3s^2+20s-32]
----
= 2(3s-4)(s+8)
=================
Cheers,
Stan H.
=================