Removing pairs of brackets
FOIL
When multiplying out a pair of brackets, multiply each term in the first bracket by each term in the second bracket.
So that this doesn't become too confusing, we use FOIL to help us.
- First: Multiply the first terms from each bracket (1st term in 1st bracket with 1st term in 2nd bracket)
- Outside: Multiply the two outside terms (1st term in 1st bracket with 2nd term in 2nd bracket)
- Inside: Multiply the two inside terms (2nd term in 1st bracket with 1st term in 2nd bracket)
- Last: Multiply the last terms from each bracket (2nd term in 1st bracket with 2nd term in 2nd bracket)
Example
Multiply and simplify \((2x + 5)(3x - 4)\)
Method 1
\((2x + 5)(3x - 4)\)
\(= (2x \times 3x) + (2x \times - 4) + (5 \times 3x) + (5 \times - 4)\)
\(= 6{x^2} - 8x + 15x - 20\)
\(= 6{x^2} + 7x - 20\)
Method 2
In this method, split the first bracket up to multiply the terms in the first bracket by the terms in the second bracket individually:
\((2x + 5)(3x - 4)\)
\(= 2x(3x - 4) + 5(3x - 4)\)
\(= 6{x^2} - 8x + 15x - 20\)
\(= 6{x^2} + 7x - 20\)
Now try the example questions below.
Question
Multiply out the following, then simplify.
\((3a+4) (2a+5)\)
\(=(3a\times 2a) + (3a\times 5) + (4\times 2a) + (4\times 5)\)
\(= 6a^{2}+15a+8a+20\)
\(=6a^{2}+23a+20\)
Question
Multiply out the following, then simplify.
\((2y - 3)(5y + 7)\)
\(= (2y \times 5y) + (2y \times 7) + ( - 3 \times 5y) + ( - 3 \times 7)\)
\(= 10{y^2} + 14y - 15y - 21\)
\(= 10{y^2} - y - 21\)
Question
Multiply out \({(2a - 5)^2}\) then simplify.
The squared outside the bracket suggests that it is multiplied by itself, therefore write this in the following way:
\(= (2a - 5)(2a - 5)\)
\(= (2a \times 2a) + (2a \times - 5) + ( - 5 \times 2a) + ( - 5 \times - 5)\)
\(= 4{a^2} - 10a - 10a + 25\)
\(= 4{a^2} - 20a + 25\)
Question
Multiply out \({(3x + 4)^2}\) and simplify.
\(= (3x + 4)(3x + 4)\)
\(= (3x \times 3x) + (3x \times 4) + (4 \times 3x) + (4 \times 4)\)
\(= 9{x^2} + 12x + 12x + 16\)
\(= 9{x^2} + 24x + 16\)
Question
Calculate the area of the rectangle below. All lengths are in cm.
Area of Rectangle = length x breadth
\(= (x + 7)(x + 4)\)
\(= (x \times x) + (x \times 4) + (7 \times x) + (7 \times 4)\)
\(= {x^2} + 4x + 7x + 28\)
\({x^2} + 11x + 28\,\)cm2
Question
Simplify the following:
\((x + 1)(3{x^2} - 4x + 1)\)
\(= (x \times 3{x^2}) + (x \times - 4x) + (x \times 1) + (1 \times 3{x^2}) + (1 \times - 4x) + (1 \times 1)\)
\(= 3{x^3} - 4{x^2} + x + 3{x^2} - 4x + 1\)
\(= 3{x^3} - {x^2} - 3x + 1\)
More guides on this topic
- Factorising an algebraic expression
- Completing the square in a quadratic expression
- Simplifying algebraic fractions
- Applying the four operations to algebraic fractions
- Determining the equation of a straight line
- Working with linear equations and inequations
- Working with simultaneous equations
- Changing the subject of a formula
- Determine the equation of a quadratic function from its graph
- Sketching a quadratic function
- Identifying features of a quadratic function
- Solving a quadratic equation
- Solving a quadratic equation using the quadratic formula
- Using the discriminant to determine the number of roots
- Video playlist