Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)

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Question: Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)

Options:

Correct Answer: -15

Solution:

The term independent of x occurs when the powers of x cancel out. The term is C(5,2)(-3)^2(1)^3 = 45.

Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)

Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)

Step 1: Identify the expression we need to expand, which is (x^2 - 3x + 1)^5.
Step 2: Understand that we are looking for the term that does not contain 'x' (the term independent of x).
Step 3: Use the multinomial expansion formula, which allows us to expand expressions like (a + b + c)^n.
Step 4: In our case, a = x^2, b = -3x, and c = 1, and n = 5.
Step 5: The general term in the expansion can be represented as C(5, k1, k2, k3) * (x^2)^k1 * (-3x)^k2 * (1)^k3, where k1 + k2 + k3 = 5.
Step 6: To find the term independent of x, we need the total power of x to be 0.
Step 7: The power of x in the term is given by 2*k1 + k2. Set this equal to 0: 2*k1 + k2 = 0.
Step 8: Since k1, k2, and k3 must be non-negative integers, the only solution is k1 = 2 and k2 = 0 (which gives k3 = 3).
Step 9: Calculate the coefficient of this term using the multinomial coefficient: C(5, 2, 0, 3).
Step 10: C(5, 2, 0, 3) = 5! / (2! * 0! * 3!) = 10.
Step 11: Now calculate the contribution from the other parts: (-3)^2 = 9 and (1)^3 = 1.
Step 12: Multiply the coefficient by the contributions: 10 * 9 * 1 = 90.
Step 13: Therefore, the term independent of x in the expansion is 90.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Finding the Term Independent of a Variable – Identifying the term in an expansion where the variable's exponent is zero, leading to a constant term.
Combinatorial Coefficients – Using binomial coefficients to determine the number of ways to choose terms from the expansion.