Problem 12 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x+3 y)^{3} $$

Step-by-Step Solution

Answer

The expanded form of \( (x+3 y)^{3} \) using the Binomial Theorem is: \(x^3 + 9x^2y + 27x*y^2 + 27y^3\)

Step 1: Expand Using The Binomial Theorem

Using the Binomial theorem, the expansion of \( (x+3 y)^{3} \) can be expressed as: \\(a + b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k\\ Here, a=x, b=3y and n=3. So, substitute these values into the formula.

Step 2: Calculate Each Term

Now, calculate each individual term in the expansion: \\- The first term is: {3 \choose 0} * (x^{3-0}) * ((3y)^0) = 1 * x^3 * 1 = x^3.\\- The second term is: {3 \choose 1} * (x^{3-1}) * ((3y)^1) = 3 * x^2 * (3y) = 9x^2y.\\- The third term is: {3 \choose 2} * (x^{3-2}) * ((3y)^2) = 3 * x * (9y^2) = 27x*y^2.\\- The fourth or last term is: {3 \choose 3} * (x^{3-3}) * ((3y)^3) = 1 * 1 * (27y^3) = 27y^3.

Step 3: Combine Terms

The final expanded form of \( (x+3 y)^{3} \) consists of all individual terms combined: \\(x^3 + 9x^2y + 27x*y^2 + 27y^3\\)