Suppose 3s represents an even integer. What polynomial represents the product of 3s, the even integer that comes just before 3s, and the even integer that comes just after 3s?
A. 27s^3 + 12s
B. 27s^3 - 3s
C. 27s^3 - 12s
D. 3s^3 - 12s
My calculations are telling me C, but I'm second guessing myself!
Even integers differ by two.
If 3s is even, the even just below 3s is (3s -2) and the even just above 3s would be (3s+2)
so
you must find product :
( 3s - 2 ) ( 3 s ) ( 3 s + 2 ) =
( 3 s * 3 s - 2 * 3 s ) ( 3 s + 2 ) =
( 9 s ^ 2 - 6 s ) ( 3 s + 2 ) =
9 s ^ 2 * 3 s + 9 s ^ 2 * 2 - 6 s * 3 s - 6 s * 2 =
27 s ^ 3 + 18 s ^ 2 - 18 s ^ 2 - 12 s =
27 s ^ 3 - 12 s
Answer C
To figure out which polynomial represents the product of 3s, the even integer that comes just before 3s, and the even integer that comes just after 3s, let's break it down step by step.
First, let's define the even integer that comes just before 3s. An even integer is any integer that is divisible by 2. Since 3s is already an even integer, the even integer that comes just before 3s would be 3s - 2.
Next, let's define the even integer that comes just after 3s. Using the same logic, it would be 3s + 2.
Now, let's multiply these three terms together:
(3s) * (3s - 2) * (3s + 2)
Expanding this expression, we get:
(9s^2 - 6s) * (3s + 2)
Multiplying these terms further, using the distributive property, we get:
27s^3 + 18s^2 - 18s^2 - 12s
Combining like terms, we end up with:
27s^3 - 12s
So the polynomial that represents the product of 3s, the even integer that comes just before 3s, and the even integer that comes just after 3s is option C: 27s^3 - 12s.
Therefore, your initial calculations were correct!
To find the polynomial that represents the product of 3s, the even integer that comes just before 3s, and the even integer that comes just after 3s, we can follow these steps:
Step 1: Let's assume the even integer before 3s is 3s - 2 and the even integer after 3s is 3s + 2.
Step 2: Now, we can find the product of these three terms:
Product = 3s * (3s - 2) * (3s + 2)
Step 3: Expanding the product using the distributive property, we get:
Product = (3s)^3 - (3s)^2 * 2 + (3s)^2 * 2 - 2^2 * 3s
Simplifying further, we have:
Product = 27s^3 - 6s^2 + 6s^2 - 12s
The terms -6s^2 and +6s^2 cancel each other out, so we are left with:
Product = 27s^3 - 12s
Therefore, the polynomial that represents the product of 3s, the even integer that comes just before 3s, and the even integer that comes just after 3s is 27s^3 - 12s.
So the correct answer is C.