IB Maths AA HL 1.9 Binomial theorem and coefficient reasoning Question Bank

Practise IB Maths AA HL 1.9 by using general binomial terms to solve parameterised coefficient and expansion problems.

Exam points

Use the general term to derive a coefficient constraint in a high-power binomial expansion.
Solve a parameterised expansion while tracking integer, real or non-zero restrictions.
Compare terms or coefficients to prove a relation and reject extraneous parameter values.

[Maximum number: 3]

Consider the expression $\frac{1}{1+x^{2}}$ , where |x|<1.

Show that the first four terms in the binomial expansion of $\frac{1}{1+x^{2}}$
are $1-x^{2}+x^{4}-x^{6}$ .

attempt to create binomial expansion to four terms with n=-1

\begin{aligned} & 1+\binom{-1}{1} x^{2}+\binom{-1}{2}\left(x^{2}\right)^{2}+\binom{-1}{3}\left(x^{2}\right)^{2}+\ldots \\ & 1-x^{2}+\frac{(-1)(-2)}{2}\left(x^{2}\right)^{2}+\frac{(-1)(-2)(-3)}{6}\left(x^{2}\right)^{3} \end{aligned}

Note: Award A1 for the third term and A1 for the fourth term.

1-x^{2}+x^{4}-x^{6}

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