Use the binomial theorem (𝑥 + 𝑦)𝑛 = 𝑥𝑛 + (𝑛 1) 𝑥𝑛−1𝑦+... + (𝑛 𝑟) 𝑥𝑛−𝑟 𝑦𝑟 +... + 𝑦𝑛 to expand expressions, e.g.(2𝑥 − 1)3 Mathematical Methods 2025 v1.3

A polynomial function is defined by \(P(x) = (3x - 2)^5\). (a) Use the binomial theorem to determine the first three terms of the expansion of \(P(x)\) in descending powers of \(x\). [2 marks] (b) Hence, determine the coefficient of \(x^3\) in the expansion of \(P(x)\). [2 marks]

Use the binomial theorem to expand $(3x + 2)^4$, simplifying your answer.

A scientist models the concentration of a chemical compound in a reaction vessel using the function \(C(t) = k(3t - 2)^6\), where \(t\) is time in hours, \(k\) is a positive constant, and \(C(t)\) is measured in micrograms per litre. (a) Use the binomial theorem to expand \((3t - 2)^6\) fully. [3 marks] (b) Hence determine the coefficient of \(t^4\) in the expansion of \(C(t)\) when \(k = 0.5\). [2 marks]

Using the binomial theorem to expand $(3x + 2)^4$, which of the following is the coefficient of the $x^3$ term?