Proofs Involving Multiplying Out Brackets and Whole Numbers

The video tutorial covers two main parts. Part A demonstrates how to expand and simplify the expression X(X-1)(X+1) to show it equals X^3 - X. Part B involves proving that the difference between a whole number and its cube is always a multiple of 6. This is done by showing that among three consecutive integers, one is always a multiple of 2 and another a multiple of 3, ensuring their product is a multiple of 6.

What is the result of expanding the expression (X-1)(X+1)?

• X^3 + X

• X^2 - 1

• X^2 + 1

• X^3 - X

In the context of the video, what is the significance of the expression X^3 - X?

• It represents a quadratic equation.

• It is used to prove a property of multiples of 6.

• It is an example of a polynomial division.

• It is unrelated to the topic discussed.

Why is it important to identify multiples of 2 and 3 in the proof?

• Because they are the only even numbers.

• Because their product is 6, which is the focus of the proof.

• Because they simplify the expression.

• Because they are prime numbers.

What is the result when you multiply two consecutive integers, one of which is a multiple of 2 and the other a multiple of 3?

• A multiple of 5

• A multiple of 6

• A multiple of 9

• A multiple of 12

Which of the following sets of numbers demonstrates the proof that the difference between a whole number and its cube is a multiple of 6?

• 4, 5, 6

• 3, 4, 5

• 2, 3, 4

• 1, 2, 3

What is the key takeaway from the proof discussed in the video?

• The difference between a number and its square is always a multiple of 6.

• The difference between a number and its cube is always a multiple of 6.

• The sum of a number and its cube is always a multiple of 6.

• The product of a number and its cube is always a multiple of 6.

What is the role of consecutive integers in the proof?

• They are used to find the average of the numbers.

• They simplify the calculation of the cube.

• They ensure that at least one number is a multiple of 2 and another is a multiple of 3.

• They help identify a pattern in the sequence.