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Why Is n³−n Always Divisible by Six?

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Exploring Algebraic Identities · Chapter challenge problems · 3:26

You'll learn: Factor into three consecutive integers and reason about divisibility.

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Why Is n³−n Always Divisible by Six?

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Why Is n³−n Always Divisible by Six?

You'll learn: Factor into three consecutive integers and reason about divisibility.

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Chapter challenge problems

3

Two Given Linear Factors Constrain a Quadratic
3:26
Two Given Linear Factors Constrain a Quadratic
You'll learn: Use the stated factors to compare coefficients and prove a relationship.
Use a Sum and Pairwise Products in a Cubic Identity
3:38
Use a Sum and Pairwise Products in a Cubic Identity
You'll learn: Express the second factor using the square-of-a-sum identity.
Evaluate Cubic Expressions from a Given Relationship
3:36
Evaluate Cubic Expressions from a Given Relationship
You'll learn: Use a three-variable cubic identity to avoid finding individual variable values.

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Chance, randomness and the probability scale

6

Probability Measures How Likely Something Is
3:50
Probability Measures How Likely Something Is
You'll learn: Introduce uncertainty and likelihood using the chapter’s everyday questions.
Know What Could Happen Without Knowing What Will Happen
3:50
Know What Could Happen Without Knowing What Will Happen
You'll learn: Explain random experiments, individual trials and possible outcomes.
Why Can a Coin Toss Be a Fair Way to Decide?
3:39
Why Can a Coin Toss Be a Fair Way to Decide?
You'll learn: Connect fairness to a symmetric coin and an unbiased random toss.
Read the Probability Scale from Zero to One
3:47
Read the Probability Scale from Zero to One
You'll learn: Place impossible, less likely, even-chance, more likely and certain events on a scale.
Fractions, Decimals and Percentages Describe the Same Chance
4:01
Fractions, Decimals and Percentages Describe the Same Chance
You'll learn: Translate between numerical forms and interpret probability statements.
Explain a Likelihood Ranking Instead of Guessing a Number
4:00
Explain a Likelihood Ranking Instead of Guessing a Number
You'll learn: Rank the chapter’s everyday events and explain the assumptions behind each judgement.

Next topic

Experimental and theoretical probability

6

Estimate Probability from What Actually Happened
4:00
Estimate Probability from What Actually Happened
You'll learn: Define experimental probability as event frequency divided by trials.
Run a Coin Experiment and Record the Results
3:49
Run a Coin Experiment and Record the Results
You'll learn: Organise a20-toss experiment and calculate observed relative frequencies.
A Paper Cup Has Three Outcomes—Are They Equally Likely?
3:53
A Paper Cup Has Three Outcomes—Are They Equally Likely?
You'll learn: Collect landing-position data and assign empirical probabilities.
Reason About a Fair Die Without Running an Experiment
3:47
Reason About a Fair Die Without Running an Experiment
You'll learn: Count favourable faces among six equally likely faces.
Count Letter Cards, Not Just Different Letters
3:52
Count Letter Cards, Not Just Different Letters
You'll learn: Use equally likely physical positions when a letter repeats.
Two Possible Results Do Not Automatically Mean Fifty-Fifty
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Two Possible Results Do Not Automatically Mean Fifty-Fifty
You'll learn: Test whether equally likely outcomes are justified.