Answer:
12 laps
Step-by-step explanation:
1/4 times 12 =3
0 | 3 8
1 | 2 2 4
2 | 0 1 3 6
3 | 4
key : 1 | 4 means 14
We're looking for a scalar function 
such that 
. That is,
Integrate the first equation with respect to 
:
Differentiate with respect to 
:
Integrate with respect to :
So 
is indeed conservative with the scalar potential function
where 
is an arbitrary constant.
Answer:
(a + b + 2c)(a² + 2ab + b² - 2ac - 2bc + 4c²)
Step-by-step explanation:
Given
(a + b)³ + 8c³ ← this is a sum of cubes and factors in general as
a³ + b³ = (a + b)(a² - ab + b²)
Thus
(a + b)³ + 8c³
= (a + b)³ + (2c)³
= (a + b + 2c)( (a + b)² - 2c(a + b) + (2c)² )
= (a + b + 2c)(a² + 2ab + b² - 2ac - 2bc + 4c²)
Multiplying positive numbers always comes out positive, no matter how many there are. So you can forget about the positive ones.
Every two negative numbers in a multiplication produce a positive.
Another way to say that is: If the number of negative ones is EVEN,
the answer is positive, and if it's ODD, the answer is negative.
That monster product of eight numbers in problem #12 will be negative,
because there are an odd amount of negative factors in it.
