Binomial expansion: แบบฝึกหัด

What is the expansion of \((1 - x)^4\) up to and including the term in \(x^2\)?

The coefficient of \(x^2\) in the expansion of \((1 + ax)^6\) is 60. Find the positive value of the constant \(a\).

In the expansion of \((2 + x)^n\), what factor must be taken out of the bracket to write it in the form \(a^n(1 + \frac{bx}{a})^n\) before expanding for non-integer \(n\)?

Find the coefficient of the \(x^3\) term in the binomial expansion of \((2 - 3x)^{-1}\) for small \(x\).

Identify the fourth term in the expansion of \((1 + x)^6\) when terms are written in ascending powers of \(x\).

State the range of values of \(x\) for which the binomial expansion of \((1 - 4x)^{\frac{1}{2}}\) is valid.

Find the first three terms in the binomial expansion of \((1 + 2x)^{-2}\), simplifying each coefficient.

Determine the coefficient of the \(x^2\) term in the binomial expansion of \((1 + 2x)^{-3}\) and state the range of values of \(x\) for which this expansion is valid.

(a) Find the first three terms, in ascending powers of \( x \), of the binomial expansion of \( (1 + kx)^8 \), where \( k \) is a non-zero constant.

(b) Given that the coefficient of \( x^2 \) in this expansion is 252, find the two possible values of \( k \).