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PSLE Foundation Mathematics

Expand \(\frac{(1-2x)^{\frac{1}{3}}}{1+x}\) up to and including the term in \(x^2\). State the common open range of validity. Explain why substituting \(x=\frac34\) is not justified by these expansions.[7]

Expand \(\frac{(1-2x)^{\frac{1}{3}}}{1+x}\) up to and including the term in \(x^2\). State the common open range of validity. Explain why substituting \(x=\frac34\) is not justified by these expansions.

General Binomial Theorem

Combining two validity conditions

Binomial Theorem

For sufficiently small \(x\), \((1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots\), where&nbsp;\(a\ne0\) and \(\alpha\in\mathbb Q\). Find \(a\), \(\alpha\), and the coefficient of \(x^3\). State the open range of validity.[7]

For sufficiently small \(x\), \((1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots\), where \(a\ne0\) and \(\alpha\in\mathbb Q\). Find \(a\), \(\alpha\), and the coefficient of \(x^3\). State the open range of validity.

Recovering a rational exponent

Expand \((1-2x)^{-3}\) up to and including the term in \(x^3\), and state the open range of validity.[5]

Expand \((1-2x)^{-3}\) up to and including the term in \(x^3\), and state the open range of validity.

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