This lesson is about expanding the cube (the third power) of a binomial expression. While you learned how to expand squares like (x+y)², this takes it one step further. The key is to memorize two main formulas and apply them carefully.

1. The Two Core Formulas

a) The Cube of a Sum: (x + y)³

The formula for expanding the cube of a sum is: (x + y)³ = x³ + 3x²y + 3xy² + y³

Let's break that down:

• x³: The cube of the first term.

• + 3x²y: Three times the square of the first term multiplied by the second term.

• + 3xy²: Three times the first term multiplied by the square of the second term.

• + y³: The cube of the second term.

Notice the pattern of coefficients: 1, 3, 3, 1 Notice the powers of x go down (3, 2, 1, 0) and the powers of y go up (0, 1, 2, 3).

Example: Expand (a + 2)³

• Here, x = a and y = 2.

• a³ + 3(a²)(2) + 3(a)(2²) + 2³

• = a³ + 6a² + 3(a)(4) + 8

• = a³ + 6a² + 12a + 8

b) The Cube of a Difference: (x - y)³

The formula for expanding the cube of a difference is very similar, but the signs alternate. (x - y)³ = x³ - 3x²y + 3xy² - y³

Notice the pattern of signs: + , - , + , -

Example: Expand (y - 4)³

• Here, x = y and y = 4.

• y³ - 3(y²)(4) + 3(y)(4²) - 4³

• = y³ - 12y² + 3(y)(16) - 64

• = y³ - 12y² + 48y - 64

2. Important Applications and Variations

Expressions with Coefficients

Remember to apply the powers to the coefficients as well as the variables.

• Example: Expand (2a - 5b)³

  • Here, x = 2a and y = 5b.

  • (2a)³ - 3(2a)²(5b) + 3(2a)(5b)² - (5b)³

  • = 8a³ - 3(4a²)(5b) + 3(2a)(25b²) - 125b³

  • = 8a³ - 60a²b + 150ab² - 125b³

Working Backwards

You might be given an expanded expression and asked to write it as a cube. Look for the pattern.

• Example: Write p³ - 9p²q + 27pq² - 27q³ as a cube.

  1. The signs are + - + -, so it's a difference (x - y)³.

  2. The first term is p³, so x = p.

  3. The last term is -27q³, which is -(3q)³, so y = 3q.

  4. Check the middle terms: -3(p)²(3q) = -9p²q (Correct). And +3(p)(3q)² = +27pq² (Correct).

  5. The answer is (p - 3q)³.

Numerical Calculations

This is a common exam question to test your understanding.

• Example: Evaluate 21³.

  1. Rewrite 21 as (20 + 1).

  2. Expand (20 + 1)³ using the formula.

  3. (20)³ + 3(20)²(1) + 3(20)(1)² + (1)³

  4. = 8000 + 3(400)(1) + 60(1) + 1

  5. = 8000 + 1200 + 60 + 1 = 9261

Exam Tips & Common Mistakes

• Mistake 1: Sign Errors. The most common mistake with (x - y)³ is getting the signs wrong. Just remember the simple alternating pattern: plus, minus, plus, minus.

• Mistake 2: Forgetting to Power the Numbers. In (2x)³, the result is 8x³, not 2x³. The power applies to everything inside the bracket.

• Mistake 3: Mixing up the Squares. Students often mix up which term to square in the middle parts. Remember 3x²y (first term squared) and 3xy² (second term squared).

• Exam Tip 1: Always write the general formula (x³ + 3x²y + ...) on your paper first, then substitute the specific terms from your problem. This helps prevent mistakes.

• Exam Tip 2: If asked to find the volume of a cube with side length (a+5), the answer is simply the expansion of (a+5)³. Don't get confused by the geometry context.