An experimental dataset suggests a relationship between variables that can be modeled using a binomial expansion. In this scenario, the form of the model is given by \((1+x)^n\) with different values of n based on the experimental conditions.
(a) For a standard case with an integer exponent, consider the model \(f(x) = (1+x)^3\). Expand \(f(x)\) completely using the binomial theorem.
(b) In a more advanced scenario, suppose the same type of model is used but with a negative exponent: \(f(x) = (1+x)^{-2}\). Using the generalized binomial theorem, write down the first three terms of the expansion.
Question
Using the model \(f(x) = (1+x)^3\), expand the expression completely using the binomial theorem.
Hint
Hint
Hint
Remember that for a positive integer n, \((1+x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + …\). Apply this to \(n=3\).
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
1 + 3x + 3x^2 + x^3
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Answer Explanation
Answer Explanation
Question
Extend your reasoning to the model \(f(x) = (1+x)^{-2}\). Using the generalized binomial theorem, write down the first three terms of the expansion.
Hint
Hint
Hint
For \( (1+x)^n \) with a negative exponent \(n\), the generalized binomial theorem is used. The first few terms are given by \(1 + nx + \frac{n(n-1)}{2!}x^2 + …\). Apply this with \(n = -2\).
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Answer
Answer
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Mark Scheme
Mark Scheme
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Explanation
Explanation
- Formula Booklet
Complete Answer
1 – 2x + 3x^2
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Mark Scheme
Mark Scheme
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Answer Explanation
Answer Explanation
