Question

Prove that 2x3 + 3x² + x is always divisible by 6 if x is an integer.

PLEASE I NEED THIS ANSWERED ASAP.
USE PROOF BY EXHAUSTION.

Answer 1

Answer: see proof below

Explanation:

Proof by exhaustion means to input several numbers that satisfy the claim and if they all prove to be true then you have proven the claim.

Claim: (2x³ + 3x² + x)/(6) ∈ Z (integer)

Let's choose x = {1, 2, 3, 4)

Case 1: x = 1

2(1)³ + 3(1)² + (1) = 12

12/6 = 2

2 ∈ Z
`\checkmark`

Case 2: x = 2

2(2)³ + 3(2)² + (2) = 30

30/6 = 5

5 ∈ Z
`\checkmark`

Case 3: x = 3

2(3)³ + 3(3)² + (3) = 84

84/6 = 14

14 ∈ Z
`\checkmark`

Case 4: x = 4

2(4)³ + 3(4)² + (4) = 180

180/6 = 30

30 ∈ Z
`\checkmark`

Since each case was shown to be true we have proven the claim is true by exhaustion.