Binomial expansion

Find, in ascending powers of x, the first four terms of the binomial expansion of

giving each term in simplest form.

Give a reason whyx = should not be used in the expansion to find an

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Find the first three terms, in ascending powers of x, of the binomial expansion of

writing each term in simplest form.

Using the answer to part (a) and using algebraic integration, estimate the value of

giving your answer to 4 significant figures.

Find, using algebraic integration, the exact value of

giving your answer in the form alnb + c, where a, b and c are constants to be found.

Find the first four terms, in ascending powers of x, of the binomial expansion of

writing each term in simplest form. A student uses this expansion with x = to find an approximation for Using the answer to part (a) and without doing any calculations,

Given that f(x) can be expressed in the form

where A, B and C are constants

find the value of B and the value of C

show that A = 0

Use binomial expansions to show that, in ascending powers of x

where p, q and r are simplified fractions to be found.

Find the range of values of x for which this expansion is valid.

Given that f(x) can be expressed in the form

where A, B and C are constants

find the value of B and the value of C

show that A = 0

Use binomial expansions to show that, in ascending powers of x

where p, q and r are simplified fractions to be found.

Find the range of values of x for which this expansion is valid.

Find the first four terms, in ascending powers of x, of the binomial expansion of

giving each term in simplest form.

Explain how you could use x = in the expansion to find an approximation for There is no need to carry out the calculation.

In the binomial expansion of

the coefficient of x⁴ is 15 120 Find the value of a.

Find the first three terms, in ascending powers of x, of the binomial expansion of

giving each coefficient in its simplest form.

Possible values of x that could be substituted into this expansion are:

Without evaluating your expansion,

state, giving a reason, which of the three values of x should not be used

Use binomial expansions to show that 1 x x²

A student substitutes x into both sides of the approximation shown in part (a) in an

Give a reason why the student should not use x

The binomial series expansion of

up to and including the term in x² is

where a and k are constants.

Find the value of a.

Find the value of k, giving your answer in its simplest form.

Hence find the numerical coefficient of x² in the series expansion of

f(x) = (2 + kx)^–3, kx < 2, where k is a positive constant The binomial expansion of f(x), in ascending powers of x, up to and including the term in x² is

where A and B are constants.

Write down the value of A.

Find the value of k.

Find the value of B.

Use the binomial series to find the expansion of

in ascending powers of x, up to and including the term in x³. Give each coefficient as a fraction in its simplest form.

Find the binomial expansion of

in ascending powers of x, up to and including the term in x². Give each coefficient in its simplest form.

Find the exact value of (4 5x when x =

Substitute x = into your binomial expansion from part (a) and hence find an

Give your answer in the form where p and q are integers.

Find the binomial expansion of

in ascending powers of x up to and including the term in x². Give each coefficient as a simplified fraction.

Hence, or otherwise, find the expansion of

in ascending powers of x, up to and including the term in x². Give each coefficient as a simplified fraction.

Given that the binomial expansion of (1 + kx)^–4, kx < 1, is

find the value of the constant k,

find the value of the constant A, giving your answer in its simplest form.

Use the binomial expansion to show that

Give your answer in the form where a and b are integers.

Find the binomial expansion of

in ascending powers ofx, up to and including the term in x°. Give each coefficient as

Use your expansion to estimate an approximate value for 7100, giving your to 4 decimal places. State the value ofx, which you use in your expansion, and show all your working. (3)

Given

find the binomial expansion of f(x), in ascending powers of x, up to and including the term in x³. Give each coefficient as a simplified fraction.

Find the binomial expansion of f (x) in ascending powers of x, up to and including the term in x³. Give each coefficient in its simplest form.

Use your answer to part (a) to find the binomial expansion in ascending powers of x, up to and including the term in x³, of

Find the first three non-zero terms of the binomial expansion of f (x) in ascending powers of x. Give each coefficient as a simplified fraction.

Use the binomial theorem to expand

in ascending powers of x, up to and including the term in x³. Give each coefficient as a simplified fraction.

f x = , x where a and b are constants.

the value of a and the value of b,

Find the values of the constants A, B and C.

term in x². Give each coefficient as a simplified fraction.

Find the binomial expansion of

in ascending powers of x up to and including the term in x³, simplifying each term.

Substitute x into the binomial expansion in part (a) and hence obtain an approximation to √23. Give your answer to 5 decimal places.

₂

Given that f (x) can be expressed in the form

find the values of B and C and show that A = 0.

Hence, or otherwise, find the series expansion of f (x), in ascending powers of x, up to and including the term in x². Simplify each term.

Find the percentage error made in using the series expansion in part (b) to estimate the value of f (0.2). Give your answer to 2 significant figures.

Use the binomial theorem to expand

simplified fraction.

Find the first 4 terms, in ascending powers of x, of the binomial expansion of

where k is a non-zero constant. Give each term in its simplest form. Given that the coefficient of x³ in this expansion is 1890

find the value of k.

Find the first 4 terms, in ascending powers of x, of the binomial expansion of

giving each term in its simplest form.

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

giving each term in its simplest form.

The expansion, in ascending powers of x, of f(x) up to and including the term in x² is

where A and B are constants.

Write down the value of A.

Find the value of k.

Hence find the value of B.

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

giving each term in its simplest form.

Hence, or otherwise, find the first 3 terms, in ascending powers of x, of the expansion of

Use the binomial theorem to find all the terms of the expansion of

Give each term in its simplest form.

Write down the expansion of

in ascending powers of x, giving each term in its simplest form.

Find the first 4 terms, in ascending powers of x, of the binomial expansion of

giving each term in its simplest form.

Find the first 3 terms, in ascending powers of x, in the binomial expansion of

Give each term in its simplest form.

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

giving each term in its simplest form.

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

where b is a non-zero constant. Give each term in its simplest form.

Given that, in this expansion, the coefficient of x² is twice the coefficient of x,

find the value of b.

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

where k is a constant. Give each term in its simplest form.

Given that the coefficient of x² is 6 times the coefficient of x,

find the value of k.

Find the first 4 terms, in ascending powers of x, of the binomial expansion of (1 + ax)¹⁰, where a is a non-zero constant. Give each term in its simplest form.

Given that, in this expansion, the coefficient of x³ is double the coefficient of x²,

find the value of a.