Exempel 1

1. $4 (x+y) = 4x + 4y$
   
   
2. $2(a-b) = 2a -2b$
   
   
3. $x \left(\displaystyle\frac{1}{x} + \displaystyle\frac{1}{x^2} \right) = x\cdot \displaystyle\frac{1}{x} + x \cdot \displaystyle\frac{1}{x^2} = \displaystyle\frac{\not{x}}{\not{x}} + \displaystyle\frac{\not{x}}{x^{\not{2}}} = 1 + \displaystyle\frac{1}{x}$
   
   
4. $a(x+y+z) = ax + ay + az$

Exempel 2

1. $-(x+y) = (-1) \cdot (x+y) = (-1)x + (-1)y = -x-y$
   
   
2. $-(x^2-x) = (-1) \cdot (x^2-x) = (-1)x^2 -(-1)x = -x^2 +x$
   där vi i sista ledet användt att $-(-1)x = (-1)(-1)x = 1\cdot x=x$
   
   
3. $-(x+y+y^3) = (-1)\cdot (x+y+y^3) = (-1)\cdot x + (-1) \cdot y -(-1)\cdot y^3$

   $=-x-y+y^3$
   
   

4. $x^2 - 2x -(3x+2) = x^2 -2x -3x-2 = x^2 -(2+3)x -2$

   $= x^2 -5x -2$
   
   

Exempel 3

1. $3x +9y = 3x + 3\cdot 3y = 3(x+3y)$
   
   
2. $xy + y^2 = xy + y\cdot y = y(x+y)$
   
   
3. $2x^2 -4x = 2x\cdot x - 2\cdot 2\cdot x = 2x(x-2)$
   
   
4. $\displaystyle \frac{y-x}{x-y} = \displaystyle \frac{-(x-y)}{x-y} = \displaystyle \frac{-1}{1} = -1$
   
   

Exempel 4

1. $(x+1)(x-2) = x\cdot x + x \cdot (-2) + 1 \cdot x + 1 \cdot (-2) = x^2 -2x+x-2$

   $=x^2 -x-2$
   
   

2. $3(x-y)(2x+1) = 3(x\cdot 2x + x\cdot 1 - y \cdot 2x - y \cdot 1) = 3(2x^2 +x-2xy-y)$

   $=6x^2 +3x-6xy-3y$
   
   

3. $(1-x)(2-x) = 1\cdot 2 + 1 \cdot (-x) -x\cdot 2 - x\cdot (-x) = 2-x-2x+x^2$

   $=2-3x+x^2$
   

   där vi använt att $-x\cdot (-x) = (-1)x \cdot (-1)x = (-1)^2 x^2 =1\cdot x^2 = x^2$.