Expansions with several variables

Expanding (a + b)^n with two variables, binomials such as (2a − 3b)^3 and (x + 1/x)^4, constant terms, and expanding products of binomial powers.

The general expansion

For example, $(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$.

Example 1●●

Expand $(2a - 3b)^3$.

$(2a)^3 + 3(2a)^2(-3b) + 3(2a)(-3b)^2 + (-3b)^3$

Row 3: 1, 3, 3, 1, with the terms 2a and −3b.

$= 8a^3 - 36a^2b + 54ab^2 - 27b^3$

Example 2●●

Expand $(2 - 3x)^4$.

$16 + 4(8)(-3x) + 6(4)(9x^2) + 4(2)(-27x^3) + 81x^4$

$= 16 - 96x + 216x^2 - 216x^3 + 81x^4$

Example 3●●

Find the term independent of $x$ in $\left(x + \dfrac1x\right)^4$.

The middle term: $6\,x^2\left(\dfrac1x\right)^2 = 6$.

The terms contain x⁴, x², x⁰, x⁻² and x⁻⁴ in turn, so only the middle one has no x.

Example 4●●●

Find the coefficient of $x^2$ in $(1 + 2x)(1 - x)^5$.

$(1 - x)^5 = 1 - 5x + 10x^2 - \cdots$; only terms up to $x^2$ matter.

$10 + 2(-5) = 0$

x² comes from 1 × 10x² and from 2x × (−5x).

Flashcards

$(a + b)^4$

$a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$

Powers in each term of $(a + b)^n$

They add to $n$.

Constant term of $left(x + rac1x ight)^4$

$6$

Practice