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How do you expand brackets with powers of 3?

Published in Algebra Expansion 1 min read

Expanding brackets with powers of 3 involves dealing with expressions like (x + y)3 or more complex variations. This process combines the binomial theorem (or Pascal's triangle) for expanding the power and the distributive property for multiplying terms. Here's how you approach it:

Understanding the Basics

Before diving into examples, it's crucial to remember the binomial expansion formula (or Pascal's Triangle) and the distributive property.

Binomial Theorem and Pascal's Triangle

The binomial theorem provides a general formula for expanding expressions of the form (a + b)n. For a power of 3, the expansion looks like this:

(a + b)3 = a3 + 3a2b + 3ab2 + b3

Pascal's Triangle offers a visual way to determine the coefficients:

      1
     1 1
    1 2 1
   1 3 3 1   <-- Coefficients for power of 3

Distributive Property

The distributive property states that a(b + c) = ab + ac. This is fundamental when multiplying expanded terms by other expressions.

Steps to Expand Brackets with Powers of 3

  1. Identify the Expression: Determine the expression you need to expand, for instance, (x + 2)3 or a more complex expression like (x + 1)(x - 2)3.

  2. Apply the Binomial Theorem (or Pascal's Triangle): If you have a simple expression like (x + 2)3, directly apply the binomial theorem:

    (x + 2)3 = x3 + 3x2(2) + 3x(2)2 + 23

  3. Simplify the Terms: Simplify each term in the expansion:

    x3 + 6x2 + 12x + 8

  4. Handle More Complex Expressions: For expressions like (x + 1)(x - 2)3:

    • First, expand (x - 2)3 using the binomial theorem:

      (x - 2)3 = x3 - 6x2 + 12x - 8

    • Then, multiply the resulting expression by the remaining bracket (x + 1):

      (x + 1)(x3 - 6x2 + 12x - 8)

      According to the reference, expand and simplify two of the brackets then multiply the resulting expression by the third bracket.

    • Use the distributive property to multiply each term in the first bracket by each term in the second bracket.

    • (x)(x3 - 6x2 + 12x - 8) + (1)(x3 - 6x2 + 12x - 8)

    • x4 - 6x3 + 12x2 - 8x + x3 - 6x2 + 12x - 8

  5. Combine Like Terms: Combine terms with the same power of x:

    x4 + (-6x3 + x3) + (12x2 - 6x2) + (-8x + 12x) - 8
    x4 - 5x3 + 6x2 + 4x - 8

Example

Let's expand (2x - 1)3:

  1. (2x - 1)3 = (2x)3 + 3(2x)2(-1) + 3(2x)(-1)2 + (-1)3
  2. Simplify: 8x3 - 12x2 + 6x - 1

Practical Tips

  • Double-Check: Always double-check your work to minimize errors, especially with signs.
  • Organization: Keep your work organized to easily track terms and combinations.
  • Practice: Practice with various examples to become comfortable with the process.

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