High School

Expand the expression \((1 - 3x)^4\).

a) \(1 - 12x + 36x^2 - 48x^3 + 27x^4\)

b) \(1 - 12x + 48x^2 - 36x^3 + 27x^4\)

c) \(1 + 12x + 36x^2 + 48x^3 + 27x^4\)

d) \(1 + 12x + 48x^2 + 36x^3 + 27x^4\)

Answer :

Final answer:

The correct expansion of (1−3x)⁴ is 1 - 12x + 54x² - 108x³ + 81x⁴ using the binomial theorem. The provided options do not include this correct expansion.

Explanation:

When expanding the expression (1−3x)⁴, we need to use the binomial theorem, which allows us to expand expressions in the form of (a+b)⁴ as

  • a⁴
  • + 4a³b
  • + 6a²b²
  • + 4ab³
  • + b⁴

In our case, a is 1 and b is -3x. Thus, we calculate each term as follows:

  1. 1⁴ = 1
  2. 4(1)³(-3x) = -12x
  3. 6(1)²(-3x)² = 6(9x²) = 54x²
  4. 4(1)(-3x)³ = -4(27x³) = -108x³
  5. (-3x)⁴ = 81x⁴

Combining these terms, we get the expanded expression: 1 - 12x + 54x² - 108x³ + 81x⁴. Therefore, the correct answer to the question is not explicitly listed in the multiple-choice options provided.

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