In the expansion of (x + 3)^5, what is the coefficient of x^3?

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Question: In the expansion of (x + 3)^5, what is the coefficient of x^3?

Options:

Correct Answer: 90

Solution:

Using the binomial theorem, the coefficient of x^3 in (x + 3)^5 is given by 5C3 * (3)^2 = 10 * 9 = 90.

In the expansion of (x + 3)^5, what is the coefficient of x^3?

In the expansion of (x + 3)^5, what is the coefficient of x^3?

Step 1: Identify the expression we are working with, which is (x + 3)^5.
Step 2: Recognize that we need to find the coefficient of x^3 in this expansion.
Step 3: Use the binomial theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 4: In our case, a = x, b = 3, and n = 5.
Step 5: We want the term where x is raised to the power of 3, which means we need k = 2 (because n - k = 3).
Step 6: Calculate the binomial coefficient 5C2, which is the number of ways to choose 2 from 5. This is calculated as 5! / (2! * (5-2)!) = 10.
Step 7: Now, calculate (3)^2, which is 9.
Step 8: Multiply the binomial coefficient by (3)^2: 10 * 9 = 90.
Step 9: Therefore, the coefficient of x^3 in the expansion of (x + 3)^5 is 90.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem, which involves combinations and powers.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion using the formula nCr * b^(n-r) where n is the exponent, r is the term number, and b is the constant.