Problem 27 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (3 x-y)^{5} $$

Step-by-Step Solution

Answer

The binomial \( (3x-y)^{5} \) expanded and simplified form with binomial theorem is \( 243x^5 - 810x^4y + 1080x^3y^2 - 720x^2y^3 + 240xy^4 - 32y^5 \)

Step 1: Identify Variables

From the given problem, we can identify that the two terms (variables) to be used are 3x and -y, while the power of the binomial is 5.

Step 2: Applying the Binomial Theorem

The binomial theorem can be stated as follows: \((a+b)^n = \sum_{k=0}^{n} {n\choose k}(a^{n-k})(b^{k}) \). Applying this theorem, the expression becomes: \((3x-y)^5 = \sum_{k=0}^{5} {5\choose k}((3x)^{5-k})((-y)^{k}) \).

Step 3: Solving Term by Term

Now we can calculate each term of the sum individually. Replace \( k \) with the values 0, 1, 2, 3, 4 to 5 consecutively. After performing these calculations and simplifying, the terms are: 243x^5, -810x^4y, 1080x^3y^2, -720x^2y^3, 240xy^4, and -32y^5.

Step 4: Form the Final Expression

Combining these terms gives the expression in simplified form: \( 243x^5 - 810x^4y + 1080x^3y^2 - 720x^2y^3 + 240xy^4 - 32y^5 \)