Expanding Brackets and Simplifying

Expanding brackets is a process that helps us simplify expressions by multiplying each term inside the brackets with the term outside.

It’s based on the distributive property, which states that when we have a number multiplied by a sum, like a(b+c), we can multiply the number by each term in the sum separately and then add the results: ab + ac.

  • Therefore, a(b + c) = ab + ac

Also: a(b + c + d + ...) = ab + ac + ad + ...

Remember that ab = a \times b.

FOIL Method

The FOIL method is a technique used to expand brackets containing two binomials, which are expressions with two terms. The acronym “FOIL” stands for “First, Outer, Inner, Last,” representing the order in which the terms in the binomials are multiplied.

This systematic approach ensures that each term in one binomial is multiplied by each term in the other binomial. Here’s how the FOIL method works:

1. First: Multiply the first terms of both binomials.

2. Outer: Multiply the outer terms of both binomials.

3. Inner: Multiply the inner terms of both binomials.

4. Last: Multiply the last terms of both binomials.

After completing these steps, combine any like terms to simplify the resulting expression.

Example:

Consider the following expression:

(A + B)(C + D)

Applying the FOIL method, we perform the following multiplications:

1. First: A × C = AC

2. Outer: A × D = AD

3. Inner: B × C = BC

4. Last: B × D = BD

Now, combine the results:

AC + AD + BC + BD

So, the expanded form of (A + B)(C + D) using the FOIL method is AC + AD + BC + BD.

Let’s look at some examples.

Examples

Example 1:

Expand and simplify the following:

i) x(x + 2)

ii) 4x(2x + 3y)

iii) 5x(4x - a + b)

iv) 3(x + 2 + y)

v) 7(x + y + xy)

i) x(x + 2) = x^2 + 2xx(x + 2) = x^2 + 2x

ii) 4x(2x + 3y) = 8x^2 + 12xy4x(2x + 3y) = 8x^2 + 12xy

iii) 5x(4x - a + b) = 20x^2 - 5ax + 5bx5x(4x - a + b) = 20x^2 - 5ax + 5bx

iv) 3(x + 2 + y) = 3x + 6 + 3y3(x + 2 + y) = 3x + 6 + 3y

v) 7(x + y + xy) = 7x + 7y + 7xy7(x + y + xy) = 7x + 7y + 7xy

Example 2:

Expand and simplify the following:

i) 3(x + y) + 2(x - y)

ii) 4(x + 2y) - (x + y)

iii) 3(x + 4) - 3(x-5)

iv) (x + 2) - (x + 3) + (3x + 5)

v) x + x(2x + 3) - 7x + 2

i) 3(x + y) + 2(x - y) = 3x + 3y + 2x - 2y3(x + y) + 2(x - y) = 3x + 3y + 2x - 2y

= 5x + y= 5x + y

ii) 4(x + 2y) - (x + y) = 4x + 8y - x - y4(x + 2y) - (x + y) = 4x + 8y - x - y

= 3x + 7y= 3x + 7y

iii) 3(x + 4) - 3(x - 5) = 3x + 12 - 3x + 153(x + 4) - 3(x - 5) = 3x + 12 - 3x + 15

= 27= 27

iv) (x + 2) - (x + 3) + (3x + 5) = x + 2 - x - 3 + 3x + 5(x + 2) - (x + 3) + (3x + 5) = x + 2 - x - 3 + 3x + 5

= 3x + 4= 3x + 4

v) x + x(2x + 3) - 7x + 2= x + 2x^2 + 3x - 7x + 2x + x(2x + 3) - 7x + 2= x + 2x^2 + 3x - 7x + 2

= 2x^2 - 3x + 2= 2x^2 - 3x + 2

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