02Cexpanding

= Expanding Brackets = toc

Single Brackets
The linear expression a(b + c) can be expanded to give ab + ac.

This is called the __distributive law__ {but you don't need to remember the name}

As with elsewhere in algebra, a pronumeral pushed up against something else is shorthand for multiply. So a(b + c) is shorthand for **a** multiplied by each of the terms enclosed in the bracket.



We can visualise the distributive law by remembering that the area of a rectangle is equal to height times length.

So if we draw a rectangle with height = a and length = b + c then Area = a(b + c)

and from the diagram (to the right) we can see that area also equals the sum of the smaller rectangles. Area = ab + ac

Hence : a(b + c) = ab + ac

** Example 1 **
math \textbf{(a)} \quad \text{Expand : } \; 7 \big( m - 4 \big) \qquad. math

math \\ . \qquad 7 \big( m - 4 \big) = 7 \times m + 7 \times -4 \qquad. \\ . \\ . \qquad \qquad \qquad = 7m - 28 math

math \textbf{(b)} \quad \text{Expand : } \; b \big( b + 7 \big) \qquad. math

math . \qquad b \big( b + 7 \big) = b^2 + 7b \qquad. math

math \textbf{(c)} \quad \text{Expand : } \; -3x \big( 4x - 5y \big) \qquad. math

math . \qquad -3x \big( 4x - 5y \big) = -12x^2 + 15xy \qquad. math

{don't forget: when __multiplying__ two negatives, you get a positive answer -- as in (c) above}

Expand and simplify
When the expression is more complicated, it is often possible to simplify by adding(or subtracting) like terms after you have expanded the brackets.

Notice that any negative signs belong with the term that comes after it, so in: math . \qquad 3 \big( a + 4 \big) - 5 \big( a - 2 \big) \qquad. math the second bracket should be multiplied by –5. (see example 2b)

** Example 2 **
math \textbf{(a)} \quad \text{Expand : } \; d \big( d - 4 \big) + 3 \big( d + 2 \big) \qquad. math

math \\ . \qquad d \big( d - 4 \big) + 3 \big( d + 2 \big) = d^2 - 4d + 3d + 6 \qquad. \\ . \\ . \qquad \qquad \qquad \qquad \qquad \;\; = d^2 -d + 6 math

math \textbf{(b)} \quad \text{Expand : } \; 3 \big( a + 4 \big) - 5 \big( a - 2 \big) \qquad. math

math \\ . \qquad 3 \big( a + 4 \big) - 5 \big( a - 2 \big) = 3a + 12 - 5a + 10 \qquad. \\ . \\ . \qquad \qquad \qquad \qquad \qquad \;\; = -2a + 22 math

** Example 4 **
Write an expression (in expanded form) for a rectangle where the length is one more than twice the width.

__**Solution:**__

Let width = w

Then length = 2w + 1

Draw a diagram of the rectangle (see right)

From the diagram, we can see that: Area = w(2w + 1)

Expand Area = 2w 2 + w

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