An expression can contain more than one variable, such as x and y. Terms are like terms only when they have the same letters with the same powers: 3xy and -5yx are like terms, but x2y and xy2 are not. Why: multiplication can be done in any order, so yx=xy and 2x x 3xy=2 x 3 x x x x x y=6x2y. Multiply the numbers, then collect each letter, adding the powers of the same letter. **Product** | **Working** | **Result** 4x x (-3y) | 4 x (-3) x x x y | -12xy 2xy2 x 5x2y | 2 x 5 x x x x2 x y2 x y | 10x3y3 -a x (-ab) | (-1) x (-1) x a x a x b | a2b Method - Multiply the term outside each bracket by every term inside, keeping the signs. - In each product, multiply the numbers and add the powers of each letter. - Write the letters of each term in alphabetical order, so yx becomes xy and like terms are easy to see. - Collect like terms. Example Expand and simplify 2a(3a-4b)-5b(a-2b). 2a(3a-4b)-5b(a-2b) = 6a2-8ab-5ab+10b2 = 6a2-13ab+10b2 Check: with a=b=1, the question gives 2(-1)-5(-1)=3 and the answer gives 6-13+10=3. Common errors: x2y and xy2 are unlike terms and cannot be added, and 3x x 4x=12x2, not 12x.

Expansion in more than one variable

Secondary 2

An expression can contain more than one variable, such as $x$ and $y$. Terms are like terms only when they have the same letters with the same powers: $3xy$ and $-5yx$ are like terms, but $x^2y$ and $xy^2$ are not.

Why: multiplication can be done in any order, so $yx=xy$ and $2x\times3xy=2\times3\times x\times x\times y=6x^2y$. Multiply the numbers, then collect each letter, adding the powers of the same letter.

Product	Working	Result
$4x\times(-3y)$	$4\times(-3)\times x\times y$	$-12xy$
$2xy^2\times5x^2y$	$2\times5\times x\times x^2\times y^2\times y$	$10x^3y^3$
$-a\times(-ab)$	$(-1)\times(-1)\times a\times a\times b$	$a^2b$

Method

Multiply the term outside each bracket by every term inside, keeping the signs.
In each product, multiply the numbers and add the powers of each letter.
Write the letters of each term in alphabetical order, so $yx$ becomes $xy$ and like terms are easy to see.
Collect like terms.

Example

Expand and simplify $2a(3a-4b)-5b(a-2b)$.

$\begin{aligned} 2a(3a-4b)-5b(a-2b) &= 6a^2-8ab-5ab+10b^2 \\ &= 6a^2-13ab+10b^2 \end{aligned}$

Check: with $a=b=1$, the question gives $2(-1)-5(-1)=3$ and the answer gives $6-13+10=3$. Common errors: $x^2y$ and $xy^2$ are unlike terms and cannot be added, and $3x\times4x=12x^2$, not $12x$.

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Product	Working	Result
\(4x\times(-3y)\)	\(4\times(-3)\times x\times y\)	\(-12xy\)
\(2xy^2\times5x^2y\)	\(2\times5\times x\times x^2\times y^2\times y\)	\(10x^3y^3\)
\(-a\times(-ab)\)	\((-1)\times(-1)\times a\times a\times b\)	\(a^2b\)