What is the coefficient of x^3 in the expansion of (x + 3)^4?

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Multiple Choice

What is the coefficient of x^3 in the expansion of (x + 3)^4?

Explanation:
In the binomial expansion, the term with x raised to a specific power comes from picking a certain number of the second addend. Here, treat (x + 3)^4 with a = x and b = 3. The x^3 term appears when the exponent of x is 3, which happens when the selection parameter k satisfies 4 − k = 3, so k = 1. The coefficient for that term is C(4,1) times 3^1, which is 4 × 3 = 12. Writing out the expansion confirms this: x^4 + 12x^3 + 54x^2 + 108x + 81, so the coefficient of x^3 is 12. The other numbers correspond to coefficients of the other powers of x in the expansion.

In the binomial expansion, the term with x raised to a specific power comes from picking a certain number of the second addend. Here, treat (x + 3)^4 with a = x and b = 3. The x^3 term appears when the exponent of x is 3, which happens when the selection parameter k satisfies 4 − k = 3, so k = 1. The coefficient for that term is C(4,1) times 3^1, which is 4 × 3 = 12. Writing out the expansion confirms this: x^4 + 12x^3 + 54x^2 + 108x + 81, so the coefficient of x^3 is 12. The other numbers correspond to coefficients of the other powers of x in the expansion.

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