Algebraic Expressions and Simplification

By the end of this lesson you will be able to:

• Name the parts of an algebraic expression
• Collect like terms to simplify expressions
• Expand and factorise single-bracket expressions

The Market Stall Variable Problem

You buy 3 mangoes. Your friend buys 5 mangoes.

We do not know the price per mango yet.

How many mangoes were bought in total?

• "3 somethings + 5 somethings = 8 somethings"
• It does not matter what a mango costs — we still have 8 mangoes

You Already Know This Idea

In patterns-03 you wrote the nth-term formula:

Here stands for any position number.

Algebra extends that: stands for any value — the unknown price per mango.

A letter is a name for a value we have not yet been given.

What a Variable Actually Is

Mangoes cost shillings each.

= variable: a letter for an unknown value. means .

Anatomy of an Algebraic Expression

Mangoes and Oranges — A Natural Constraint

Mangoes cost shillings; oranges cost shillings.

• 3 mangoes:
• 5 oranges:
• Together:

Can become one term?

No — like-terms rule: only terms with the same variable part combine.

Like Terms and Unlike Terms

Same variable part → combine. Different → stay separate.

Collecting Like Terms — Worked Examples

Rule: Add/subtract coefficients; keep the variable.

1. and share variable ; has no partner
3. -term stays: →

Probe: Spot the Collecting Error

A student writes: . What went wrong?

A. Unlike terms merged ( and )
B. Variable dropped
C. absorbed into the -group

Correct answer: — errors A and C: was merged into the -group.

Check-In: Collect Like Terms Practice

Simplify. Show your working.

Answers:

Bundle Deal — A New Problem

3 bundles, each = mango price + 4-shs handling fee.

Total:

How do we find the total in terms of ?

Open the bracket — multiply 3 into every term inside. This is expanding brackets.

The Distributive Law — Numeric Recall

The 3 multiplies every term. Same with a variable:

General form:

Expanding Brackets — Worked Examples

Sign: , not .

The arrows show both multiplications. The multiplier reaches every term inside.

Watch Out: Partial Expansion Error

Error: multiply only the first term.

Check :

• Wrong: ✗
• Correct: ✓

Expanding with a Variable Multiplier

• (x-squared)

and are unlike terms — do not combine them.

Same distributive law: the multiplier reaches every term inside.

Check-In: Expand the Brackets Practice

Expand each expression. Show every multiplication step.

(Work in your exercise book.)

Answers:

Factorising Is Expanding in Reverse

The same fact, read in opposite directions. Factorising is expanding reversed.

How to Factorise — The HCF Method

Pull out the HCF of all terms. Steps: find HCF → write outside bracket → divide each term → verify.

Factorise :

• HCF
• ;
• ✓

Factorising — Two Worked Examples

: HCF → ✓

: HCF → ✓

Verify each by expanding back.

Mixed Simplification — Expand Then Collect

Simplify:

Expand:

Collect: ; has no partner.

Result:

Always expand before collecting.

Your Turn: Full Mixed Problem

Simplify:

Show full working.

What Algebra Lets You Do First

A variable names the unknown. An expression reasons before the value is known.

• — true for any mango price
• Expand, collect, factorise — all before a single number appears
• In algebra-02: substitute values to turn expressions into answers