Learning Objectives
  • Translate verbal statements into algebraic expressions.
  • Substitute values into formulas and expressions.
  • Collect like terms and expand brackets.
  • Factorise expressions by taking out a common factor.
Key Terms
Variable
A letter representing a number that may change.
Constant
A fixed numerical value.
Coefficient
The numerical multiplier of a variable.
Term
A number, variable or product separated from other terms by addition or subtraction.
Like terms
Terms with exactly the same variable part and powers.
Expression
A mathematical phrase without an equality sign.
Factorise
Rewrite an expression as a product of factors.
Writing Algebraic Expressions

Letters represent general numbers. Algebraic notation is compact: 5x means 5\times x, and ab means a\times b. The multiplication sign is usually omitted between a number and a variable or between variables.

Words Algebraic expression
Five more than x x+5
Five less than x x-5
Five less than twice x 2x-5
The square of the sum of x and 3 (x+3)^2
The sum of the squares of x and 3 x^2+3^2
The mean of a, b and c \frac{a+b+c}{3}

Order matters. “Subtract x from 10” means 10-x, not x-10. Brackets are essential when an operation applies to a whole expression.

Substitution

Replace every variable by its given value, using brackets around negative values. Apply the correct order of operations.

Worked Example: Substitution

Question: Evaluate 3a^2-2ab+b when a=-2 and b=5.

  1. Substitute using brackets: 3(-2)^2-2(-2)(5)+5.
  2. Calculate powers first: 3(4)+20+5.
  3. Add: 12+20+5=37.

Answer: 37

Collecting Like Terms

Only like terms can be combined. The variable part and index must match exactly. For example, 4x+7x=11x, but 4x+7x^2 cannot be simplified.

Worked Example: Simplifying

Question: Simplify 5a-3b+2a+8b-4.

  1. Combine a-terms: 5a+2a=7a.
  2. Combine b-terms: -3b+8b=5b.
  3. The constant remains -4.

Answer: 7a+5b-4

Expanding Brackets

Multiply every term inside a bracket by the factor outside. With two brackets, multiply every term in the first bracket by every term in the second.

a(b+c)=ab+ac
Worked Example: Single Bracket

Question: Expand and simplify 4(3x-2)-5(x+1).

  1. Expand: 12x-8-5x-5.
  2. Collect like terms.

Answer: 7x-13

Worked Example: Two Brackets

Question: Expand (2x-3)(x+5).

  1. Multiply 2x by both terms: 2x^2+10x.
  2. Multiply -3 by both terms: -3x-15.
  3. Collect like terms.

Answer: 2x^2+7x-15

Factorising By A Common Factor

Factorising reverses expansion. Identify the greatest factor common to every term and place it outside a bracket.

Worked Example: Common Factor

Question: Factorise completely 18x^2y-12xy^2.

  1. The HCF of 18 and 12 is 6.
  2. Both terms contain x and y; use the smallest powers x^1y^1.
  3. Divide each term by 6xy.

Answer: 6xy(3x-2y)

Using Algebra In Perimeter Problems

Algebraic expressions often describe lengths. Add all sides for perimeter, and simplify. If a shape has equal sides, use multiplication rather than writing repeated addition.

Worked Example: Perimeter

Question: A rectangle has length 3x+2 cm and width x-1 cm. Find and simplify its perimeter.

  1. Use P=2l+2w.
  2. Substitute: P=2(3x+2)+2(x-1).
  3. Expand and simplify: 6x+4+2x-2=8x+2.

Answer: (8x+2)\text{ cm}.

8th Edition Chapter Map
  • Basic algebraic concepts and notation
  • Addition and subtraction of linear terms
  • Expansion and factorisation of linear expressions
  • Linear expressions with fractional coefficients
Translating Words Into Algebra

Algebraic notation compresses a relationship. Order matters: “five less than x” is x-5, while “five minus x” is 5-x. “The square of the sum” (a+b)^2 is not the same as “the sum of the squares” a^2+b^2.

Words Expression
three more than twice n 2n+3
half the difference between p and 7 \frac{p-7}{2}
the total cost of x items at 240 each plus delivery 500 240x+500
the perimeter of a rectangle of sides a and b 2a+2b
Substitution With Negative Values

Put substituted negative numbers in brackets. If a=-3, then a^2=(-3)^2=9, whereas -a^2=-9. Follow the order of operations after substitution.

Worked Example: Substitution

Evaluate 3x^2-2xy+y when x=-2 and y=5.

3(-2)^2-2(-2)(5)+5=12+20+5=37.

Collecting, Expanding And Factorising

Only like terms can be collected. Terms are like when their variable parts, including powers, are identical. Expansion removes brackets by multiplying every term inside. Factorisation reverses expansion by extracting a common factor.

Worked Example: Multiple Operations

Simplify 4(2x-3)-3(x+5)+2x.

8x-12-3x-15+2x=7x-27.

Worked Example: Factorising Completely

Factorise 18a^2b-12ab^2+6ab. The greatest common factor is 6ab, giving 6ab(3a-2b+1).

Fractional Coefficients And Algebraic Fractions

Fractional coefficients follow the same rules as numerical fractions. Use a common denominator when adding coefficients. Simple algebraic fractions can often be simplified by factorising the numerator and denominator and cancelling common factors, subject to restrictions on values that make an original denominator zero.

Worked Example: Fractional Linear Expression

Simplify \frac34x-\frac56x+\frac23x. The coefficient is \frac{9-10+8}{12}=\frac7{12}, so the result is \frac7{12}x.

Worked Example: Simplifying A Rational Expression

\frac{6x^2+12x}{3x}=\frac{6x(x+2)}{3x}=2(x+2), where x\ne0.

Extended Practice

A. Simplify 5a-2(3a-4)+7.

5a-6a+8+7=15-a.

B. Factorise 15xy-25x^2y completely.

5xy(3-5x).

C. Write an expression for the mean of a,b,c and d.

\frac{a+b+c+d}{4}.

Examination Guidance
  • Use brackets whenever substituting a negative number.
  • Keep expressions exact; do not insert an equals sign unless making a true equation.
  • Check factorisation by expanding your answer.
  • When translating words, identify which operation happens first and use brackets accordingly.
Common Mistakes
  • Combining unlike terms such as 3x+2x^2=5x^3.
  • Writing (a+b)^2=a^2+b^2; the middle term is missing.
  • Changing signs incorrectly when expanding a negative bracket.
  • Factorising with a common factor that is not the greatest when the question says “factorise completely”.
Knowledge Check

1. Write an expression for “three less than twice n”.

Answer

2n-3.

2. Evaluate 2x^2-3y when x=-3 and y=4.

Answer

6.

3. Simplify 7p-4q-3p+9q.

Answer

4p+5q.

4. Expand (x+4)(x-7).

Answer

x^2-3x-28.

5. Factorise completely 15a^2b+20ab^2.

Answer

5ab(3a+4b).

Mixed Review With Full Solutions

1. Simplify 3(2a-5)-2(4-a)+7.

6a-15-8+2a+7=8a-16.

2. Factorise 24x^2y-18xy^2+6xy completely.

The greatest common factor is 6xy, giving 6xy(4x-3y+1).

3. Evaluate \frac{2p-q}{3} when p=-4.5 and q=6.

(2(-4.5)-6)/3=(-9-6)/3=-5.

4. Simplify \frac23x-\frac58x+\frac14x.

Coefficient =\frac{16-15+6}{24}=\frac7{24}, so the expression is \frac7{24}x.

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