Expanding (1-2x + 3x^2)^3 using the Binomial Theorem:

The Binomial Theorem is a powerful tool used to expand expressions of the form (a + b)^n, where a and b are any real numbers and n is a positive integer. In this case, we are asked to expand the expression (1-2x + 3x^2)^3.

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer n, the expansion of (a + b)^n can be expressed as the sum of the terms:

C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n) * a^0 * b^n

where C(n, r) represents the binomial coefficient, which is calculated using the formula:

C(n, r) = n! / (r! * (n-r)!)

where n! is the factorial of n.

Step 2: Determine the values of a, b, and n

In our case, a = 1, b = -2x, and n = 3.

Step 3: Apply the Binomial Theorem

Using the formula for the binomial coefficient and the given values, we can expand the expression (1-2x + 3x^2)^3 as follows:

(1-2x + 3x^2)^3 = C(3, 0) * 1^3 * (-2x)^0 + C(3, 1) * 1^2 * (-2x)^1 + C(3, 2) * 1^1 * (-2x)^2 + C(3, 3) * 1^0 * (-2x)^3

Simplifying each term:

= C(3, 0) * 1 * 1 + C(3, 1) * 1 * (-2x) + C(3, 2) * 1 * (4x^2) + C(3, 3) * 1 * (-8x^3)

= 1 - 6x + 12x^2 - 8x^3

Step 4: Finalize the expanded expression

Therefore, the expanded form of (1-2x + 3x^2)^3 is 1 - 6x + 12x^2 - 8x^3.

Summary:

Using the Binomial Theorem, we were able to expand the expression (1-2x + 3x^2)^3. By substituting the values of a, b, and n into the formula and simplifying each term, we obtained the final expanded form of 1 - 6x + 12x^2 - 8x^3. The Binomial Theorem is a useful tool for expanding binomial expressions and allows us to easily calculate higher powers of binomials.