Algebraic Expressions
Algebra · Algebraic Expressions
Syllabus tag: KCSE | Mathematics | Form 1 | Topic 10 Algebraic Expressions
Lesson objectives
By the end of this topic, you should be able to:
- Write statements in algebraic form.
- Simplify algebraic expressions by collecting like terms.
- Factorise algebraic expressions by grouping.
- Expand and simplify expressions involving brackets.
Algebraic Expressions
An expression is a combination of letters, numbers and operations, with no equals sign.
a) Writing statements in algebra
Name each unknown and state what it represents.
"Five more than twice a number" becomes 2n + 5.
"The cost of x pens at 20 shillings each" becomes 20x.
Order matters in subtraction and division. "Three less than y" is y − 3, not 3 − y.
b) Terms and coefficients
A term is a part separated by + or −.
The coefficient is the number in front of a letter.
A constant has no letter at all.
In 5x² − 3x + 7, there are three terms, coefficients 5 and −3, and the constant 7.
The sign in front belongs to the term that follows it.
c) Collecting like terms
Like terms have identical letters raised to identical powers.
So 3x and 5x are like terms. But 3x and 3x² are not, and neither are 3x and 3y.
Add or subtract the coefficients and leave the letter part unchanged.
So 4x + 3y − x + 2y simplifies to 3x + 5y.
d) Expanding brackets
Multiply every term inside the bracket by the term outside.
So 3(2x − 5) becomes 6x − 15.
Take care with a negative outside. −2(x − 4) becomes −2x + 8, because minus times minus is plus.
e) Expanding two brackets
Every term in the first bracket multiplies every term in the second. Four products in total.
Some people remember this as FOIL: First, Outer, Inner, Last.
Then collect the like terms in the middle.
f) Factorising by common factors
Factorising reverses expanding.
Find the highest common factor of all the terms and take it outside.
So 6x + 9 becomes 3(2x + 3), and 4xy + 6x becomes 2x(2y + 3).
Always check by expanding back.
g) Factorising by grouping
Some expressions with four terms have no factor common to all of them.
Group the terms in pairs. Factorise each pair separately.
If the same bracket appears in both, take it out as a common factor.
The pairing must be chosen so the repeated bracket appears. If it does not, try pairing the terms differently.
h) Where this is used
Every formula is an expression. Rearranging formulas, solving equations and simplifying before substituting all begin here.
Words to know
- Term -- a part of an expression separated by + or − signs.
- Coefficient -- the numerical part of a term.
- Like terms -- terms with exactly the same letters raised to the same indices.
- Expand -- to remove brackets by multiplying out.
- Factorise -- to write an expression as a product, reversing expansion.
:::checkpoint Check yourself
- Simplify 7a + 4b − 2a + b.
- Expand and simplify (x + 2)(x + 6).
- Factorise 12m + 18mn.
- Factorise by grouping: 2x + 2y + ax + ay. :::
Bridge to practice
The exercises begin with writing statements algebraically and identifying terms, move through collecting like terms, expanding and factorising, and finish with grouping and substitution. After every factorisation, expand your answer mentally to confirm it reproduces the original.