Key points about simplifying expressions and expanding brackets
- To simplify an An expression is a set of terms combined using the operations +, β, π₯ or Γ·. For example 5π₯2 β 3π₯π¦ + 17. An expression does not have an equals sign., group An element within an algebraic sentence, eg π₯ or 5π¦ or 3πΒ². Elements (terms) are separated by + or β signs. together by collecting, multiplying or dividing like terms.
- To expand an expression that contains one bracket, multiply everything inside the bracket by the letter or number outside the bracket.
- To expand an expression containing two brackets, multiply every term in the first bracket by every term in the second bracket.
Refresh your knowledge on the order of operations and negative numbers to help with your understanding of algebra.
Video β Expanding double brackets
Watch this video to see how to expand double brackets using the grid or FOIL method.
Expanding double brackets.
Let's look at two different methods for expanding double brackets.
Question 1 asks you to expand the brackets 2π₯ add 8 multiplied by π₯ subtract 3. Let's use the grid method for this one.
In a grid like this, write the terms from one bracket in the top row and the terms from the otherbracket down the left-hand column. Then, fill in the middle boxes by multiplying each term from the top row with each term from the left-hand column. 2π₯ multiplied by π₯ equals 2π₯ squared, 2π₯ multiplied by β3 equals β6, 8 multiplied by π₯ equals 8π₯, and 8 multiplied by β3 equals -24.
Then, adding these values together gives the expansion of this set of brackets: 2π₯ squared add 8π₯ subtract 6π₯ subtract 24. Finally, simplify where you can. 8 and β6π₯ are both terms in π₯, so they can be simplified. Be careful with negative numbers here. 8π₯ subtract 6π₯ equals 2π₯. So, the final answer is 2π₯ squared add 2π₯ subtract 24.
OK. Let's look at another question: expand the brackets π₯ add 2 multiplied by π₯ subtract 5. Let's use the FOIL method for this one. FOIL stands for: First, Outer, Inner, and Last. This gives you an order for multiplying the terms in the first bracket by those in the second bracket.
So, start by multiplying the first term in each bracket: π₯ multiplied by π₯ equals π₯ squared. Next, multiply the two outer terms, which are the terms on the outside of the expression: π₯ multiplied by β5 equals β5π₯. Then multiply the two inner terms. These are the two on the inside of the expression. 2 multiplied by π₯ equals 2π₯. Finally, multiply the last term in each bracket. These are 2 and β5, which multiply to give β10.
Always remember to simplify your answer by collecting like terms. β5π₯ and 2π₯ are both π₯ terms. So they can be simplified to give β3π₯. This gives a final answer of π₯ squared subtract 3π₯ subtract 10.
Either the grid method or FOIL can be used to expand double brackets, so choose the one you're most confident with when answering these types of questions.
Algebraic definitions
| Key term | Definition |
|---|---|
| Variable | A letter that represents an unknown value. The letters π₯ or π¦ are often used as variables. The value of a variable can change. |
| Term | A number or letter on its own, or numbers and letters multiplied together, such as β2, 3π₯ or 5πΒ². |
| Expression | One or more terms combined using the operations +, β, Γ or Γ·. For example β4π₯ β 3 or 5π₯Β² β 3π₯π¦ + 17. An expression does not have an equals sign. |
| Equation | An equation states that two expressions are equal in value, and contains an equals sign. For example 4π β 2 = 6. An equation is only true for certain values. |
| Identity | A statement that is true no matter what values are chosen, for example 4π Γ πΒ² β‘ 4πΒ³. The triple equals sign (β‘) means βis always equal toβ. |
| Formula | A rule that links two or more variables. For example π£ = π’ + ππ‘. |
Writing expressions
Statements can be written as An expression is a set of terms combined using the operations +, β, π₯ or Γ·. For example 5π₯2 β 3π₯π¦ + 17. An expression does not have an equals sign. using algebra.
Follow the working out below
GCSE exam-style questions
- John is π years old.
Kim is three years younger than John.
Vanessa is half Kim's age.
Write an expression for each person's age.
Kim is three years younger than John, so Kim is π β 3 years old.
Vanessa is half Kim's age, so take Kim's age and divide by 2.
This gives Vanessa's age as \(\frac{(π -3)}{2}\).
- The number of oranges Naledi has is π and the number of grapes she has is π. The number of oranges she has is half the number of grapes.
Which of these statements is correct: π = 2π or π = 2π?
π = 2π
The number of grapes, π, is twice the number of oranges, π.
Check the answer with an example:
Suppose Naledi has 12 grapes, so π = 12.
She has half as many oranges as she has grapes, so π = 6.
In this example, when π = 6, π = 12. π is twice as much as π.
We can write this as an equation: π = 2π.
Simplifying expressions
Expressions where terms are added or subtracted can be simplified by collecting An element within an algebraic sentence, eg π₯ or 5π¦ or 3πΒ². Elements (terms) are separated by + or β signs. that contain the same An unknown value, usually represented by a letter such as π₯ or π¦. . These terms are called 'like terms.'
In the expression 5π + 2πΒ² + 3π β 6π + πΒ², the terms 5π and +3π are like terms. 2πΒ² and πΒ² are also like terms. β6π does not have a like term.
Expressions where terms are multiplied or divided can sometimes be simplified by using Indices are powers. For example, 3 to the power of 2, written 3Β². The singular for indices is 'index'..
πΒ² Γ πΒ³ means π Γ π Γ π Γ π Γ π, which is written as πβ΅.
When multiplying terms with indices, add the powers.πβ΅ Γ· πΒ³ = πΒ².
When dividing terms with indices, subtract the powers.
Follow the working out below
GCSE exam-style questions
- Simplify 2π₯ β 4π¦ + 1 β 3π₯ + 7π¦ β 4.
β π₯ + 3π¦ β 3
Collect the like terms:
- 2π₯ β 3π₯ = βπ₯
- β 4π¦ + 7π¦ = 3π¦
- +1 β 4 = β3
- Simplify 3ππΒ²πΒ³ Γ 4πΒ²πΒ³πβ΄.
12πΒ³πβ΅πβ·
Multiply all the terms that involve the same letter by adding the indices.
For example, πΒ² x πΒ³ = π x π x π x π x π = πβ΅
Expanding a single bracket
Expanding a brackets means multiplying everything inside the bracket by the letter or number outside the bracket.
For example, to Expanding a brackets means multiplying everything inside the bracket by the letter or number outside the bracket to remove the brackets, eg 3(π + 7) = 3 Γ π + 3 Γ 7 = 3π + 21. the An expression is a set of terms combined using the operations +, β, π₯ or Γ·. For example 5π₯2 β 3π₯π¦ + 17. An expression does not have an equals sign. 3(π + 7), both π and 7 must be multiplied by 3:
3(π + 7) = 3 Γ π + 3 Γ 7 = 3π + 21.
The process of expanding does not change the value of the expression. This means that 3(π + 7) and 3π + 21 are The same as but in a different form, eg Β½ is equivalent to 50%. to each other.
Expanding brackets uses the skills of simplifying algebra.
Follow the working out below
GCSE exam-style questions
- Expand 5πΒ³π(4ππ β 2πΒ²πΒ³).
20πβ΄πΒ² β 10πβ΅πβ΄
- Multiply both terms in the bracket by 5πΒ³π.
- Simplify the two expressions.
- Remember to add the powers when multiplying terms with indices.
- Expand and simplify 8π + 2π(3π + 7).
6πΒ² + 22π
- Remember to use the correct order of operations.
- Expand the bracket first:
- 2π(3π + 7) = 6πΒ² + 14π
- Add like terms. 8π and 14π are like terms so can be added to give 22π.
- Expand and simplify 3(2π + 5) β 4(π - 5).
2c + 35
- Expand the first bracket:
- 3(2c + 5) = 6c + 15
- Expand the second bracket. Remember β4 Γ β5 = 20:
- 4(c β 5) = β 4c + 20
- Collect like terms:
- 6c β 4c = 2c
- 15 + 20 = 35
Game - Simplifying expressions
Complete these simplifying expressions questions from our Divided Islands game.
Play the full Divided Islands game.
Expanding double brackets
Writing two brackets next to each other means the brackets need to be multiplied together.
For example (π¦ + 2)(π¦ + 3) means (π¦ + 2) Γ (π¦ + 3).
When expanding double brackets, every term in the first bracket has to be multiplied by every term in the second bracket.
Follow the working out below
GCSE exam-style questions
- Expand (3π₯ β 2)(2π₯ β 4).
6π₯Β² β 16π₯ + 8
- Multiply every term in the first bracket by every term in the second bracket.
- Remember 3π₯ Γ 2π₯ means 3 Γ π₯ Γ 2 Γ π₯.
- 3π₯ Γ 2π₯ = 6π₯Β²
- β2 Γ β4 = 8
- Simplify the like terms β12π₯ and β4π₯ to get β16π₯.
- Expand (3π₯ β 2)(2π₯ + 1).
6π₯Β² β π₯ β 2
- If using a grid, write the terms from the brackets on the side and top.
- Fill in the inside of the grid by multiplying the terms together.
- Add the four terms together and collect the like terms.
Quiz - Expanding brackets
Practise what you've learned about simplifying expressions and expanding brackets with this quiz.
Higher - How to expand three brackets
To expand three brackets, first expand and simplify two of the brackets, then multiply the result by the remaining bracket.
Follow the working out below
GCSE exam-style questions
Expand (π₯ β 2)(π₯ + 6)(π₯ β 3).
π₯Β² + π₯Β³ β 24π₯ + 36
- βββExpand the first two brackets:
- (π₯ β 2)(π₯ + 6) = π₯Β² + 6π₯ β 2π₯ β 12.
βββCollect like terms to give π₯Β² + 4π₯ β 12.
βββMultiply π₯Β² + 4π₯ β 12 by π₯ β 3. Multiply every term in the first bracket by every term in the second bracket.ββ
βββCollect like terms to give the simplified answer.ββ
Higher - Quiz - Expanding three brackets
Practise what you've learned about simplifying expressions and expanding three brackets with this quiz.
Now you've revised simplifying expressions and expanding brackets, why not look at algebraic reasoning and proof?
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