02Cexpandingdouble

= Expanding Brackets = toc

Double Brackets
When two brackets are pushed together like this : (a + b)(x + y) it means each term in the first bracket (a and b) is multiplied by each term in the second bracket, so: We can visualise this as a rectangle with a height of (a + b) and a length of (x + y) The area of the rectangle is therefore given by: Area = (a + b)(x + y)

From the diagram (right) we can see that the area is also the sum of the smaller rectangles: Area = ax + ay +bx + by

Hence : (a + b)(x + y) = ax + ay + bx + by

** Example 3a **
math \textbf{(a)} \quad \text{Expand : } \; \big( x + 5 \big)\big( x - 2\big) math



FOIL
FOIL is a shortcut for expanding double brackets.

If you consistently follow the same order ( **F** irst, **O** utside, **I** nside, **L** ast), you are less likely to make mistakes by missing out a term.
 * 1) ** First ** : Multiply the first terms from each bracket
 * 2) ** Outside ** : Mutliply the two outside terms (#1 times #4)
 * 3) ** Inside ** : Multiply the inside terms (#2 times #3)
 * 4) ** Last ** : Multiply the last terms from each bracket

{If you squint, the lines drawn in green form a happy face}

** Example 3b **
math \textbf{(b)} \quad \text{Expand : } \; \big( x - 3 \big)\big( 2x + 1 \big) \qquad \text{using FOIL} math




 * Example 4 **

math . \qquad \text{Expand : } \; 3\big( 2x + 4 \big) \big( 3x - 2 \big) math


 * Solution:**

math \\ . \qquad 3\big( 2x + 4 \big) \big( 3x - 2 \big) \qquad \qquad \{ \textit{use FOIL on the two brackets} \} \\. \\ . \qquad = 3 \big( 6x^2 - 4x + 12x - 8 \big) \qquad \{ \textit{now simplify by collecting like terms} \} \\. \\ . \qquad = 3 \big( 6x^2 + 8x - 8 \big) \qquad \qquad \{ \textit{now expand the outer bracket} \} \\. \\ . \qquad = 18x^2 + 24x - 24 math

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