Find the term independent of x in the expansion of (x^2 - 2x + 3)^4. (2022)

Find the term independent of x in the expansion of (x^2 - 2x + 3)^4. (2022)

Step 1: Identify the expression to expand, which is (x^2 - 2x + 3)^4.
Step 2: Use the multinomial expansion formula to expand the expression. The general term in the expansion is given by: T = (4! / (a! b! c!)) * (x^2)^a * (-2x)^b * (3)^c, where a + b + c = 4.
Step 3: Determine the values of a, b, and c that will make the term independent of x. This means the total power of x in the term must equal 0.
Step 4: The power of x in the term is given by 2a + b. Set this equal to 0: 2a + b = 0.
Step 5: Since a + b + c = 4, we can express c in terms of a and b: c = 4 - a - b.
Step 6: Substitute b = -2a into the equation a + b + c = 4: a - 2a + (4 - a - (-2a)) = 4, which simplifies to 4 = 4, confirming our values are consistent.
Step 7: Choose a = 0, b = 0, c = 4, which gives us the term: T = (4! / (0! 0! 4!)) * (x^2)^0 * (-2x)^0 * (3)^4.
Step 8: Calculate the term: T = 1 * 1 * 1 * 81 = 81.
Step 9: Therefore, the term independent of x in the expansion is 81.

Binomial Expansion – Understanding how to expand polynomials using the binomial theorem.
Finding the Constant Term – Identifying the term in the expansion that does not contain the variable x.
Combining Like Terms – Recognizing how different terms contribute to the overall power of x.