Question

Find P(5)-p(3) if P(2x)+P(4x)+P(6x)=24x-6

Answer 1

Answer:

P(5) - P(3) = 4

Step-by-step 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