S1

{
   "voice_prompt": "",
   "manuscript": {
       "title": {
           "text": "How to Use Pascal’s Triangle for Binomial Expansions?",
           "audio": "How to use Pascal’s Triangle for binomial expansions?"
       },
       "description": {
           "text": "When expanding a binomial expression like (a + b)ⁿ, the coefficients in the expansion correspond to the rows of Pascal’s Triangle.",
           "audio": "When expanding a binomial expression like a plus b to the power of n, the coefficients in the expansion correspond to the rows of Pascal’s Triangle."
       },
       "scenes": [
           {
               "text": "Pascal’s Triangle gives you the coefficients you need when expanding expressions like a plus b to a power of n.",
               "latex": "",
               "//": "Show Pascal's Triangle here (first 5 rows)"
           },
           {
               "text": "Expanding means writing out all the terms. Binomial, just means an expression with two terms, like a plus b.",
               "latex": "a + b"
           },
           {
               "text": "Let's see how you actually use Pascal's Triangle when expanding a binomial. Start with: a plus b squared.",
               "latex": "",
               "//": "Show Pascal's Triangle and highlight row 2"
           },
           {
               "text": "For example, when you expand a plus b squared, you get a squared plus two a b plus b squared. The coefficients are one, two, and one.",
               "latex": "(a + b)^2 = a^2 + 2ab + b^2"
           },
           {
               "text": "Notice how those coefficients match row two in Pascal’s Triangle. Also notice that you start at 0 when counting the rows. This pattern continues for any power.",
               "latex": ""
           },
           {
               "text": "When you use the coefficients, start with the first term having the highest power and decrease it step by step, while the second term starts at zero and increases step by step.",
               "latex": "",
               "//": "Show a + b and how powers shift from a^2 b^0, to a^1 b^1, to a^0 b^2 while matching the coefficients"
           },
           {
               "text": "That gives you one times a squared, plus two times a times b, plus one times b squared. This matches the triangle exactly.",
               "latex": "1a^2 + 2ab + 1b^2"
           },
           {
               "text": "You usually skip writing the ones, so the final answer is just a squared plus two a b plus b squared.",
               "latex": "a^2 + 2ab + b^2"
           },
           {
               "text": "When expanding a plus b to the power of four, use row four from the triangle. The coefficients are one, four, six, four, and one.",
               "latex": "(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4"
           },
           {
               "text": "If your binomial has something other than a and b, just follow the same pattern. Use the coefficients, and adjust the powers of each term.",
               "latex": ""
           },
           {
               "text": "For example, if you expand x plus one to the power of four, use the same row and multiply carefully term by term. That gives you one times x to the fourth times one, plus four times x cubed times one, plus six times x squared times one squared, plus four times x times one cubed, plus one times one to the fourth.",
               "latex": "(x + 1)^4 = 1x^4 \cdot 1 + 4x^3 \cdot 1 + 6x^2 \cdot 1^2 + 4x \cdot 1^3 + 1^4"
           },
           {
               "text": "You usually skip writing the ones, so the final answer is x to the fourth plus four x cubed plus six x squared plus four x plus one.",
               "latex": "x^4 + 4x^3 + 6x^2 + 4x + 1"
           }
       ],
       "outro": {
              "text": "When expanding a binomial expression like (a + b)ⁿ, the coefficients in the expansion correspond to the rows of Pascal’s Triangle.",
           "audio": "When expanding a binomial expression like a plus b to the power of n, the coefficients in the expansion correspond to the rows of Pascal’s Triangle."
       }
   }
}

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