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Expanding brackets

Expanding brackets

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Explainer Video

Tutor: Toby

Summary

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

1.
Choose the pair of brackets you are going to expand first.
2.
Once you have decided the pair of brackets, multiply them. Note: Be careful with the signs!
3.
Simplify by grouping terms.
4.
Repeat steps 1-3 until you reach the final answer. When you finish, everything has to be simplified to the maximum. 


Example 1

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

​3x(6x−4y)=18x2−12xy3x(6x-4y) = 18x^2 - 12xy3x(6x−4y)=18x2−12xy​​


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)-4x(6y-x)−4x(6y−x). Note: Be careful with the minus sign! 

​−4x(6y−x)=−24xy+4x2- 4x(6y-x) = -24xy + 4x^2−4x(6y−x)=−24xy+4x2​​

​​

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

(x−4)(y−5)=xy−5x−4y+20(x-4)(y-5) = xy -5x -4y +20(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)=18x2−12xy−24xy+4x2+xy−5x−4y+203x(6x-4y) - 4x(6y-x) + (x-4)(y-5) = 18x^2-12xy-24xy+4x^2+xy-5x-4y+203x(6x−4y)−4x(6y−x)+(x−4)(y−5)=18x2−12xy−24xy+4x2+xy−5x−4y+20


Therefore 3x(6x−4y)−4x(6y−x)+(x−4)(y−5)=22x2−35xy−5x−4y+20‾\underline{3x(6x-4y) - 4x(6y-x) + (x-4)(y-5) = 22x^2 - 35xy -5x - 4y+20}3x(6x−4y)−4x(6y−x)+(x−4)(y−5)=22x2−35xy−5x−4y+20​



Example 2

Expand and simplify (5x2+7xy)(3y−x)(5x^2 + 7xy)(3y-x)(5x2+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.

(5x2+7xy)(3y−x)=15x2y−5x3+21xy2−7x2y(5x^2 + 7xy)(3y-x) = 15x^2 y - 5x^3 + 21 xy^2 -7x^2y(5x2+7xy)(3y−x)=15x2y−5x3+21xy2−7x2y​​

​

Simplify by grouping terms: 

5x2y−5x3+21xy2−7x2y=8x2y−5x3+21xy25x^2 y - 5x^3 + 21 xy^2 -7x^2y = 8x^2 y - 5x^3 + 21 xy^25x2y−5x3+21xy2−7x2y=8x2y−5x3+21xy2​​


Therefore (5x2+7xy)(3y−x)=8x2y−5x3+21xy2‾\underline{(5x^2 + 7xy)(3y-x) = 8 x^2 y -5x^3 + 21 xy^2}(5x2+7xy)(3y−x)=8x2y−5x3+21xy2​


Example 3

Expand and simplify (x+2)(2x−3)(x+5)(x+2)(2x-3)(x+5)(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:

(x2−3x+4x−6)(x+5) (x2+x−6)(x+5)(x^2-3x+4x-6)(x+5)\\\space\\(x^2+x-6)(x+5)(x2−3x+4x−6)(x+5) (x2+x−6)(x+5)​​


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

x3+5x2+x2+5x−6x−30 underlinex3+6x2−x−30x^3+5x^2+x^2+5x-6x-30\\\space\\underline{x^3+6x^2-x-30}x3+5x2+x2+5x−6x−30 underlinex3+6x2−x−30​​

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Single brackets: Expanding and factorising

Unit 1

Single brackets: Expanding and factorising

Expanding double brackets: FOIL and multiplication grid

Unit 2

Expanding double brackets: FOIL and multiplication grid

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Expanding brackets

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Expanding brackets

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FAQs - Frequently Asked Questions

Which bracket is solved first?

When multiplying various brackets, you will have to choose where to start. When decided, you will have to follow the steps of expanding brackets.

Why do you expand brackets?

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.

What is the rule of expanding brackets?

To expand brackets you have to multiply each term in one bracket by each term in the other bracket.

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