The binomial theorem

Warm-up

Three quick questions from earlier lessons.

Discover

$(1 + x)^2 = 1 + 2x + x^2$ and $(1 + x)^3 = 1 + 3x + 3x^2 + x^3$ . The coefficients $1, 3, 3, 1$ are a row of Pascal's triangle. What is the next row?

$1, 4, 6, 4, 1$
$1, 4, 4, 4, 1$

Key idea

The binomial theorem expands any power of a sum.

(a + b)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} b^{\,k}

, with

\dbinom{n}{k} = \dfrac{n!}{k!\,(n - k)!}

. For example

\dbinom{n}{1} = n

and

\dbinom{n}{2} = \dfrac{n(n - 1)}{2}

.

Worked example

Find the coefficient of $x^2$ in $(x + 3)^5$ .

Term:

\dbinom{5}{3} x^2 \cdot 3^3

x² means b = 3 appears 3 times: k = 3 with b = 3.

10 \cdot 27 = 270

C(5, 3) = 10.

Your turn, guided

Your turn: the coefficient of $x^2$ in $(1 + x)^6$ .

\dbinom{6}{2} = \dfrac{6 \cdot 5}{2}

C(6, 2).

= 15

30 ÷ 2.

Practice ★

Find the coefficient of $x^2$ in $(1 + x)^{ 10 }$ .
Find the coefficient of $x^2$ in $(1 + x)^{ 7 }$ .
Find the coefficient of $x^2$ in $(1 + x)^{ 5 }$ .

Watch out!

Practice ★★

Find the coefficient of $x$ in $(x + 4)^{ 4 }$ .
Find the coefficient of $x$ in $(x + 3)^{ 5 }$ .
Find the coefficient of $x$ in $(x + 5)^{ 4 }$ .

In $(2x + 2)^2 = 4x^2 + kx + 4$ , what is $k$ ?
In $(7x + 2)^2 = 49x^2 + kx + 4$ , what is $k$ ?

Exit ticket

Three questions to check you have got it.

Find the coefficient of $x$ in $(x + 4)^{ 4 }$ .
Find the coefficient of $x$ in $(x + 3)^{ 5 }$ .
Find the coefficient of $x$ in $(x + 5)^{ 4 }$ .

Summary card

The binomial theorem

$(a + b)^n = \sum \dbinom{n}{k} a^{\,n-k} b^{\,k}$ .
Coefficients: a row of Pascal's triangle.
Include the powers of the constants.

Watch out: $(a + b)^2 \ne a^2 + b^2$ .

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