If the Binomial Theorem is used to expand (3x - 2)^4, what is the constant term?

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Question: If the Binomial Theorem is used to expand (3x - 2)^4, what is the constant term?

Options:

Correct Answer: -81

Solution:

The constant term occurs when x^0, which is the term with k = 4: C(4, 4)(-2)^4 = 16, thus the constant term is -16.

If the Binomial Theorem is used to expand (3x - 2)^4, what is the constant term?

If the Binomial Theorem is used to expand (3x - 2)^4, what is the constant term?

Step 1: Identify the expression to expand, which is (3x - 2)^4.
Step 2: Recall the Binomial Theorem, which states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = 3x, b = -2, and n = 4.
Step 4: We need to find the constant term, which occurs when the power of x is 0 (x^0).
Step 5: Set the exponent of x to 0: 3x raised to the power of (4 - k) = 0, which means 4 - k = 0, so k = 4.
Step 6: Calculate the term for k = 4 using the formula: C(4, 4) * (3x)^(4-4) * (-2)^4.
Step 7: C(4, 4) = 1, (3x)^0 = 1, and (-2)^4 = 16.
Step 8: Multiply these values together: 1 * 1 * 16 = 16.
Step 9: The constant term is 16.

Binomial Theorem – A formula that provides a way to expand expressions of the form (a + b)^n.
Constant Term – The term in a polynomial that does not contain any variables, specifically when the variable's exponent is zero.
Combinatorial Coefficient – The coefficient C(n, k) represents the number of ways to choose k elements from a set of n elements.