Answer:
P(5) - P(3) = 4
Explanation:
Lets explain how to solve the problem
Assume that P(x) is a linear function, that because the sum of P(2x),
P(4x), and P(6x) is linear ⇒ (24x - 6 is linear)
∵ The form of the linear function is y = ax + b
∴ P(x) = ax + b
Substitute x by 2x
∵ P(2x) = a(2x) + b
∴ P(2x) = 2ax + b
Substitute x by 4x
∵ P(4x) = a(4x) + b
∴ P(4x) = 4ax + b
Substitute x by 6x
∵ P(6x) = a(6x) + b
∴ P(6x) = 6ax + b
Add the three functions
∴ P(2x) + P(4x) + P(6x) = 2ax + b + 4ax + b + 6ax + b
Add like terms
∴ P(2x) + P(4x) + P(6x) = 12ax + 3b ⇒ (1)
∵ P(2x) + P(4x) + P(6x) = 24x - 6 ⇒ (2)
Equate (1) and (2)
∴ 12ax + 3b = 24x - 6
By comparing the two sides
∴ 12a = 24 and 3b = -6
∵ 12a = 24
Divide both sides by 12
∴ a = 2
∵ 3b = -6
Divide both sides by 3
∴ b = -2
Substitute these values in P(x)
∵ P(x) = ax + b
∴ P(x) = 2x + (-2)
∴ P(x) = 2x - 2
Now we can find P(5) - P(3)
∵ P(5) = 2(5) - 2 = 10 - 2 = 8
∵ P(3) = 2(3) - 2 = 6 - 2 = 4
∴ P(5) - P(3) = 8 - 4 = 4
* P(5) - P(3) = 4