Expanding brackets

In a nutshell

You may be asked to find the product of various terms in various algebraic expressions. It is therefore important to know how to expand brackets. To expand brackets, you have to multiply each term in one bracket by each term in the other bracket. 

You expand brackets in order to expand the expression, have an overview of the complete expression and then be able to simplify it if possible.

Expanding brackets

​​Procedure

Example 1

Expand and simplify $$3x(6x-4y) - 4x(6y-x) + (x-4)(y-5)$$

Start by choosing where you are going to operate first. In this case, start off with $$3x(6x-4y)$$:

​$$3x(6x-4y) = 18x^2 - 12xy$$​​

You can't simplify anything yet, so go back to the first step and choose another term to expand. This time, expand $$-4x(6y-x)$$. Note: Be careful with the minus sign! 

​$$- 4x(6y-x) = -24xy + 4x^2$$​​

​​

Go back again to the first step and operate in the last pair of brackets, $$(x-4)(y-5)$$​: 

$$(x-4)(y-5) = xy -5x -4y +20$$​​

Simplify the whole expression by grouping terms. 

​$$3x(6x-4y) - 4x(6y-x) + (x-4)(y-5) = 18x^2-12xy-24xy+4x^2+xy-5x-4y+20$$

Therefore $$\underline{3x(6x-4y) - 4x(6y-x) + (x-4)(y-5) = 22x^2 - 35xy -5x - 4y+20}$$

Example 2

Expand and simplify $$(5x^2 + 7xy)(3y-x)$$

There are only two brackets, so you do not need to choose where to start in this example. Multiply the brackets, and be careful with the signs.

$$(5x^2 + 7xy)(3y-x) = 15x^2 y - 5x^3 + 21 xy^2 -7x^2y$$​​

​

Simplify by grouping terms: 

$$5x^2 y - 5x^3 + 21 xy^2 -7x^2y = 8x^2 y - 5x^3 + 21 xy^2$$​​

Therefore $$\underline{(5x^2 + 7xy)(3y-x) = 8 x^2 y -5x^3 + 21 xy^2}$$

Example 3

Expand and simplify $$(x+2)(2x-3)(x+5)$$

Pick a pair of brackets to start with. You can expand the first two first. Multiply each term in one by each term in the other:

$$(x^2-3x+4x-6)(x+5)\\\space\\(x^2+x-6)(x+5)$$​​

Now expand these brackets. Again, multiply each term in one set of brackets by each term in the other:

$$x^3+5x^2+x^2+5x-6x-30\\\space\\underline{x^3+6x^2-x-30}$$​​