6x³ — Coefficients: 2 × 3 = 6. Exponents add: 2 + 1 = 3. So the answer is 6x³.

8x − 4y — Multiply both terms inside the brackets by 4: 4 × 2x − 4 × y = 8x − 4y.

Algebraic Expressions

Q: Simplify 3x + 2y − x + 5y.

2x + 7y — Group the like terms: (3x − x) + (2y + 5y) = 2x + 7y.

Q: Work out (4a − 3b) + (−2a + 5b).

2a + 2b — (4a − 2a) + (−3b + 5b) = 2a + 2b.

Q: Work out (3x + 2y) − (x − 4y).

2x + 6y — Flip the sign of every term you are subtracting: 3x + 2y − x + 4y = 2x + 6y.

Q: Work out 12a²b ÷ 3a.

4ab — 12 ÷ 3 = 4 and a² ÷ a = a, which leaves 4ab.

Q: When a = 2 and b = −1, what is the value of 3a − 2b?

8 — 3(2) − 2(−1) = 6 + 2 = 8. Subtracting a negative adds.

Q: You buy 5 items costing ¥a each and 3 items costing ¥b each. Which expression gives the total?

5a + 3b — 5 items × ¥a plus 3 items × ¥b gives 5a + 3b yen.

Where this is used

Revenue = price × quantity. Distance = speed × time. Letters let you state a rule once and use it for every set of numbers. Once you can add, subtract, multiply and divide those expressions, budgets, physics simulations and spreadsheet formulas all open up. Handling several variables at once is what turns a messy business question into a line you can actually solve.

Key points

Monomials and Polynomials

A monomial is an expression built only by multiplying numbers and letters together. Add monomials and you get a polynomial. Each monomial inside it is a term, and a term with no letter in it is the constant term.

Combining Like Terms

Terms with exactly the same letter part are like terms. Combine them by adding or subtracting their coefficients — the letter part stays as it is.

Adding and Subtracting Polynomials

To add two polynomials, just collect the like terms. To subtract, flip the sign of every term in the expression being subtracted, then add as usual.

Multiplying and Dividing Monomials

Multiply (or divide) the coefficients, then handle the letters with the exponent rule: a^m × a^n = a^{m+n}.

Costing a Shopping Basket

Buy x items at ¥a each and y items at ¥b each and the total is ax + by yen. That single polynomial covers every basket you could possibly fill.

Check yourself 10questions

Tap an option to check

Q1 Simplify 3x + 2y − x + 5y.

Q2 Work out (4a − 3b) + (−2a + 5b).

Q3 Work out (3x + 2y) − (x − 4y).

Q4 Work out 2x² × 3x.

Q5 Work out 12a²b ÷ 3a.

Q6 When a = 2 and b = −1, what is the value of 3a − 2b?

Q7 You buy 5 items costing ¥a each and 3 items costing ¥b each. Which expression gives the total?

Q8 Expand 4(2x − y).

Q9 What is the degree of the polynomial 3x²y + 2xy − 5?

Q10 Work out (5x − 3) − 2(x + 1).

Jobs that use this

Accountant / Tax Adviser $110k

Software Engineer $130k

Marketing Analyst $95k

Accountant / Tax Adviser$110k

Models revenue, costs and tax as expressions, then builds financial plans and tax strategies on top of them.

Maths used: Algebraic expressionsEquationsStatisticsCompound interest

Software Engineer$130k

Variables, expressions and assignment are the basics of programming. Complex logic gets built up one expression at a time.

Maths used: Algebraic expressionsBoolean logicAlgorithmsLinear algebra

Marketing Analyst$95k

Builds multi-variable business formulas — ad spend × conversion rate = customers won — and tunes them to squeeze out the best return.

Maths used: Algebraic expressionsStatisticsRegression analysisOptimization