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Question: Use Pascal's Triangle to expand and simplify (-3x + 5)4.

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Sure, I can guide you through the process. First, remember that the fourth line of Pascal's Triangle is 1, 4, 6, 4, 1. When you use the Binomial Theorem, the formula (a+b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n, the coefficients (or multipliers) are the same as in the corresponding line of Pascal's Triangle. Here, a = -3x, b = 5, and n = 4. Remember that "n choose k" is equivalent to the entries in Pascal's Triangle. Now plug these values in to get the following terms: - For k = 0, the term is (4 choose 0)*(-3x)^(4-0)*5^0 = 1*(-3x)^4*1 = 81x^4. - For k = 1, the term is (4 choose 1)*(-3x)^(4-1)*5^1 = 4*(-3x)^3*5 = -540x^3. - For k = 2, the term is (4 choose 2)*(-3x)^(4-2)*5^2 = 6*(-3x)^2*25 = 1350x^2. - For k = 3, the term is (4 choose 3)*(-3x)^(4-3)*5^3 = 4*(-3x)*125 = -1500x. - For k = 4, the term is (4 choose 4)*(-3x)^(4-4)*5^4 = 1*625 = 625. When you add these together, the expanded and simplified form of (-3x + 5)^4 is 81x^4 - 540x^3 + 1350x^2 - 1500x + 625.

July 24, 2023, 8:56 a.m.

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This article was generated in part by one of OpenAI’s large-scale language-generation model, answering user daoxlol192's question. Images were generated with OpenAI's AI model, DALL·E 2. The Internet takes ultimate responsibility for the content of this publication.