Expanding and Simplifying Expressions
This topic covers the essential rules for manipulating algebraic expressions. Mastering these skills is crucial as they are the building blocks for solving equations, simplifying complex formulae, and tackling other advanced maths topics in the ESAT.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Only 'like terms' can be added or subtracted. Like terms must have the exact same variables raised to the identical powers (e.g., `4xy2` and `-xy2` are like terms, but `4x2y` is not).
- Expanding brackets involves multiplying every term inside the bracket by the term outside it. For binomials like `(a+b)(c+d)`, every term in the first bracket multiplies every term in the second.
- Factorising is the reverse of expanding. It involves identifying the highest common factor (HCF) of all terms in an expression and 'pulling it out' of the bracket.
- When simplifying, always expand all brackets first, then proceed to collect all the like terms. Be systematic to avoid missing terms or signs.
- The order of terms in a final simplified expression does not matter, but conventionally they are written in descending powers of a variable (e.g., `3x3 + x2 - 5x + 1`).
Formulae
a(b + c) = ab + ac To expand a single term over a bracket (the distributive law).
(a + b)(c + d) = ac + ad + bc + bd To expand the product of two binomials. Often remembered by the acronym FOIL (First, Outer, Inner, Last).
(x + a)(x - a) = x2 - a2 A special case for expanding or factorising called the 'difference of two squares'.
(x + a)2 = x2 + 2ax + a2 A special case for expanding a 'perfect square'.
Definitions
- Like Terms
- Terms that share the same variable part, including identical powers. For example, `5x2` and `-2x2` are like terms.
- Binomial
- An algebraic expression that consists of exactly two terms, such as `(x + 3)` or `(2y - 7)`.
- Factorise
- To express an algebraic expression as a product of its factors. For example, factorising `4x + 8` gives `4(x + 2)`.
- Expand
- To multiply out the terms within brackets to remove them. For example, expanding `3(x - 5)` gives `3x - 15`.
Worked example
Expand and simplify the expression: `(2x - 5)(x + 3) - 3x(x - 4)`.
- 1
First, expand the product of the two binomials:
`(2x - 5)(x + 3) = 2x(x) + 2x(3) - 5(x) - 5(3) = 2x2 + 6x - 5x - 15`.
- 2
Simplify this part by collecting like terms:
`2x2 + x - 15`.
- 3
Next, expand the second part of the expression, making sure to multiply by `-3x`, not just `3x`:
`-3x(x - 4) = -3x(x) -3x(-4) = -3x2 + 12x` - 4
Now, combine the two simplified parts:
`(2x2 + x - 15) + (-3x2 + 12x)`.
- 5
Finally, collect all like terms from the combined expression:
`(2x2 - 3x2) + (x + 12x) - 15`.
- 6
The final simplified answer is `-x2 + 13x - 15`.
Answer: -x2 + 13x - 15
Common mistakes
- ×Sign errors are the most common mistake. When a minus sign is outside a bracket, e.g., `-(4x - 5)`, it must be applied to every term inside, giving `-4x + 5`.
- ×Forgetting that multiplying two negative numbers results in a positive one, e.g., in `(x - 2)(x - 6)`, the final term is `(-2) × (-6) = +12`.
- ×Incorrectly combining unlike terms. You cannot simplify `x2 + x` any further. Check that both the variable and its power are identical before adding or subtracting.
- ×When factorising, only taking out a partial factor instead of the highest common factor. For `12x2 + 18x`, the HCF is `6x`, not `2x` or `3`.
No-calculator tips
- ✓Use a systematic approach like FOIL (First, Outer, Inner, Last) for expanding two binomials to ensure no term is missed.
- ✓When finding a numerical highest common factor, mentally list the factors of the smallest number and check which is the largest that also divides the other numbers.
- ✓To check your final answer, substitute a simple value like `x=1` or `x=2` into both the original expression and your simplified answer. If they produce different numerical results, you've made a mistake.