CAIE A-Level Mathematics 1.6.1 Binomial expansion

Practise expanding (a + b)^n, selecting powers of x, and using binomial coefficients to find constants, coefficients or parameters.

[Maximum number: 6]

(a)

Give the complete expansion of $\left(x+\frac{2}{x}\right)^{5}$ .

[ 2 ]

$x^{5}+10 x^{3}+40 x+\frac{80}{x}+\frac{80}{x^{3}}+\frac{32}{x^{5}}$ or $x^{5}+10 x^{3}+40 x+80 x^{-1}+80 x^{-3}+32 x^{-5}$

B2, 1, 0

B2, all terms correct, B1 5 terms correct. Terms must be simplified. Lists of terms allowed.

(b)

In the expansion of $\left(a+b x^{2}\right)\left(x+\frac{2}{x}\right)^{5}$ , the coefficient of x is zero and the coefficient of $\frac{1}{x}$ is 80 . Find the values of the constants a and b.

[ 4 ]

their $40 \times a+\left(\right.$ their coefficient of $\left.x^{-1}\right) \times b=0$

M1

Coefficients of a and b must be non-zero, allow x 's so
long as they are dealt with correctly.

(their coefficient of $\left.x^{-1}\right) \times a+\left(\right.$ their coefficient of $\left.x^{-3}\right) \times b=80$

M1

Coefficients of a and b must be non-zero, allow x 's as
long as they are dealt with correctly.

$a=2 \quad b=-1$

A1 A1

Dependent on both M marks, may be seen without working.

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1.6.1—Binomial expansion question 1

Question (a)

Question (b)