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Use Pascal's Triangle to expand . Express your answer in simplest form.Answer:
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Q. Use Pascal's Triangle to expand . Express your answer in simplest form.Answer:
- Identify Row for Exponent : Identify the row of Pascal's Triangle that corresponds to the exponent . The th row (since we start counting from the th row for the exponent ) of Pascal's Triangle is . These numbers will be the coefficients in the expanded form.
- Write Expansion Using Binomial Theorem: Write out the terms of the expansion using the binomial theorem and the coefficients from Pascal's Triangle.The binomial theorem states that , where denotes the sum over from to . For , and , so the expansion will be:
- Simplify Each Term: Simplify each term in the expansion.Now we simplify each term by calculating the powers and multiplying the coefficients:This simplifies to:
- Check for Simplification: Check for any possible simplification or combination of like terms. There are no like terms to combine, and each term is already simplified.
- Write Final Answer: Write the final answer in standard polynomial form, ordering the terms from highest degree to lowest degree.The final expanded form of is:
