D1PureStretch

Binomial Expansion for Rational Powers

Q: When is a binomial expansion finite?

Only when n is a non-negative integer. Then there are exactly n + 1 terms.

Q: Why does the rational-power expansion need |x| < 1?

The series is infinite, and it only converges — adds up to a finite value equal to (1+x)^n — when the powers of x shrink, which needs |x| < 1.

Q: Should I use Pascal’s triangle or the formula?

Pascal’s triangle is quick for small n. For n above about 6, or for a single coefficient, use nr.

Q: How is this connected to probability?

nr counts the ways to get r successes in n trials, which is why it appears in the binomial distribution.

Expanding $(2 + x)^{2}$ by hand is quick. Expanding $(2 + x)^{9}$ by hand is miserable. The $\textbf{binomial theorem}$ writes down any term of $(a + b)^{n}$ directly, and a second version extends the idea to powers like $-1$ and $\tfrac12$ , where the expansion never stops.

45 min Sequences and Series

The big picture

Every term in $(a + b)^{n}$ comes from choosing either $a$ or $b$ from each of the $n$ brackets. The coefficient of $a^{n-r}b^{r}$ is simply the number of ways to choose which $r$ brackets supply the $b$ — that is $\binom{n}{r}$ , the same number that counts selections in probability and appears in Pascal’s triangle. So the theorem is really a counting argument. For positive integer $n$ the expansion is finite; for negative or fractional $n$ the coefficients never reach zero, the series is infinite, and it only converges for small enough $x$ . Those infinite expansions are how calculators approximate roots, and they reappear in the binomial distribution in statistics.

What you'll be able to do

Use Pascal’s triangle and $\binom{n}{r} = \dfrac{n!}{r!(n-r)!}$ for binomial coefficients
Expand $(a + bx)^{n}$ for positive integer $n$ and find specific terms
Expand $(1 + x)^{n}$ for rational $n$ and state the range of validity
Use binomial expansions for approximations

Binomial coefficients

The coefficients in $(a + b)^{n}$ form row $n$ of $\textbf{Pascal’s triangle}$ , where each entry is the sum of the two above it. Row 4 is $1, 4, 6, 4, 1$ .

For larger $n$ , use the formula $\binom{n}{r} = {}^{n}\mathrm{C}_{r} = \dfrac{n!}{r!(n-r)!}$ , where $n! = n \times (n-1) \times \dots \times 1$ and $0! = 1$ .

The coefficients are symmetric: $\binom{n}{r} = \binom{n}{n-r}$ , because choosing $r$ brackets to supply $b$ is the same as choosing $n - r$ to supply $a$ .

\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}

Number of ways to choose r from n.

Why this works

Calculating $\binom{10}{3}$ by the formula, most of $10!$ cancels with $7!$ : $\dfrac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$ . Always cancel before multiplying.

Positive integer powers

For positive integer $n$ : $(a + b)^{n} = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r}$ . There are $n + 1$ terms; the powers of $a$ fall from $n$ to $0$ as the powers of $b$ rise from $0$ to $n$ .

With $(a + bx)^{n}$ , the " $b$ " in the theorem is the whole of $bx$ . Put it in brackets so that its coefficient is raised to the power too.

The general term is $\binom{n}{r} a^{n-r}(bx)^{r}$ , which lets you find one term without expanding everything.

(a + b)^{n} = a^{n} + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^{2} + \dots + b^{n}

The binomial theorem, n a positive integer.

Example 1Find the first three terms, in ascending powers of

x

, of

(2 - 3x)^{6}

r = 0

2^{6} = 64

r = 1

\binom{6}{1} 2^{5}(-3x) = 6 \times 32 \times (-3x) = -576x

r = 2

\binom{6}{2} 2^{4}(-3x)^{2} = 15 \times 16 \times 9x^{2} = 2160x^{2}

Answer

64 - 576x + 2160x^{2}

Example 2Find the coefficient of

x^{4}

\left(x + \dfrac{2}{x}\right)^{8}

1General term:

\binom{8}{r} x^{8-r}\left(\dfrac{2}{x}\right)^{r} = \binom{8}{r} 2^{r} x^{8-2r}

2Need

8 - 2r = 4

, so

r = 2

3Coefficient:

\binom{8}{2} \times 2^{2} = 28 \times 4

Answer

112

Tip — Negative signs inside the bracket are the biggest source of lost marks. Write $(-3x)^{2}$ and $(-3x)^{3}$ explicitly so the sign of each term is automatic.

Rational powers

When rational $n$ is negative or is not an integer, the binomial expansion of $(1+x)^n$ is an infinite series: the coefficients do not terminate. For a non-negative integer $n$ , the expansion terminates and is valid for every real $x$ .

For the non-terminating expansions in this course, use the standard validity range $|x|<1$ . Do not assume convergence at either endpoint without a separate justification; the usual exam validity statement uses the strict inequality.

For $(a + bx)^{n}$ , first take out a factor of $a^{n}$ : $(a + bx)^{n} = a^{n}\left(1 + \dfrac{b}{a}x\right)^{n}$ , which is valid for $\left|\dfrac{bx}{a}\right| < 1$ , that is $|x| < \left|\dfrac{a}{b}\right|$ .

(1 + x)^{n} = 1 + nx + \dfrac{n(n-1)}{2!}x^{2} + \dfrac{n(n-1)(n-2)}{3!}x^{3} + \dots, \quad |x| < 1

The general binomial expansion.

Example 1Expand

\sqrt{4 + x}

up to and including the term in

x^{2}

, stating the range of validity.

\sqrt{4 + x} = 2\left(1 + \dfrac{x}{4}\right)^{1/2}

2With

n = \tfrac12

and

\dfrac{x}{4}

in place of

x

1 + \tfrac12\left(\dfrac{x}{4}\right) + \dfrac{\tfrac12\left(-\tfrac12\right)}{2}\left(\dfrac{x}{4}\right)^{2}

= 1 + \dfrac{x}{8} - \dfrac{x^{2}}{128}

4Multiply by 2. Valid when

\left|\dfrac{x}{4}\right| < 1

Answer

2 + \dfrac{x}{4} - \dfrac{x^{2}}{64}

, valid for

|x| < 4

Why this works

For a positive integer $n$ , the factor $(n - n)$ eventually appears in every coefficient and all later terms vanish. For any other $n$ that factor never arrives — which is exactly why the series goes on forever.

Approximations

For small $x$ , the higher powers are tiny, so the first few terms of an expansion give a good approximation.

To approximate a number, choose $x$ so that the bracket matches it, and check that the value lies inside the range of validity.

Example 1Use the expansion of

\sqrt{4 + x}

above to estimate

\sqrt{4.2}

1Set

x = 0.2

, which satisfies

|x| < 4

2 + \dfrac{0.2}{4} - \dfrac{0.04}{64} = 2 + 0.05 - 0.000625

Answer

2.049375

(the true value is

2.049390\dots

)

Tip — The smaller $x$ is, the better the approximation. Say why an approximation is good: " $x$ is small, so the omitted terms in $x^{3}$ and beyond are negligible."

Think like an examiner

Keep $(bx)^{r}$ bracketed, including negative signs.
Use the general term to find a single coefficient instead of expanding fully.
For rational powers, take out $a^{n}$ before expanding and state the range of validity.
When approximating, show that the chosen $x$ lies within the range of validity.

Common misconceptions

Writing the

x^{2}

term of

(1 + 2x)^{5}

10 \times 2x^{2} = 20x^{2}

Raise the whole term:

(2x)^{2} = 4x^{2}

, so the term is

40x^{2}

Expanding

(4 + x)^{1/2}

4^{1/2}(1 + x)^{1/2}

Factor out correctly:

2\left(1 + \tfrac{x}{4}\right)^{1/2}

Stating validity as

|x| < 1

for

(1 + 3x)^{-1}

Need

|3x| < 1

, so

|x| < \tfrac13

Forgetting that

0! = 1

\binom{n}{0} = \binom{n}{n} = 1

Binomial expansion

\binom{n}{r} = \dfrac{n!}{r!(n-r)!}

Binomial coefficient.

\binom{n}{r} a^{n-r}(bx)^{r}

General term.

(1+x)^{n} = 1 + nx + \tfrac{n(n-1)}{2!}x^{2} + \dots

Rational n,

|x| < 1

Stretch yourself

In the expansion of $(1 + kx)^{n}$ , where $n$ is a positive integer, the coefficient of $x$ is $12$ and the coefficient of $x^{2}$ is $60$ . Find $n$ and $k$ .

Hint — Write both coefficients in terms of $n$ and $k$ , then divide one equation by the square of the other to eliminate $k$ .

Questions students ask

When is a binomial expansion finite?

Only when

n

is a non-negative integer. Then there are exactly

n + 1

terms.

Why does the rational-power expansion need

|x| < 1

The series is infinite, and it only converges — adds up to a finite value equal to

(1+x)^{n}

— when the powers of

x

shrink, which needs

|x| < 1

Should I use Pascal’s triangle or the formula?

Pascal’s triangle is quick for small

n

. For

n

above about 6, or for a single coefficient, use

\binom{n}{r}

How is this connected to probability?

\binom{n}{r}

counts the ways to get

r

successes in

n

trials, which is why it appears in the binomial distribution.

Key takeaways

Coefficients are $\binom{n}{r}$ , from Pascal’s triangle or $\dfrac{n!}{r!(n-r)!}$ .
For positive integer $n$ , $(a + b)^{n}$ has $n + 1$ terms with general term $\binom{n}{r}a^{n-r}b^{r}$ .
For rational $n$ , $(1 + x)^{n}$ is an infinite series valid for $|x| < 1$ .
For $(a + bx)^{n}$ with rational $n$ , factor out $a^{n}$ ; validity is $|x| < |a/b|$ .
The first few terms give approximations when $x$ is small.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

All 8 lessons

Binomial Expansion for Rational Powers

Expanding

(2 + x)^{2}

by hand is quick. Expanding

(2 + x)^{9}

by hand is miserable. The

\textbf{binomial theorem}

writes down any term of

(a + b)^{n}

directly, and a second version extends the idea to powers like

-1

and

\tfrac12

, where the expansion never stops.

45 min Sequences and Series