Algebraic Expressions
Learn the vocabulary first, then the three main identities, then three factoring methods.
A. Key Terms
Variable
A letter that stands for an unknown number. It can take different values.
Examplex, y, n, a are all variables.
Constant
A fixed number — it never changes and has no letter attached.
Example5, −3, 100 are constants. They stay the same always.
Term
A single piece of an expression — can be a number, a variable, or both multiplied together.
ExampleIn 3x² + 5y − 7: the terms are 3x², 5y, and −7 (three separate terms).
Coefficient
The number multiplied in front of the variable in a term.
ExampleIn 4x², the coefficient is 4. In −7y, the coefficient is −7.
Like Terms
Terms with the same variable and same power. Only like terms can be added or subtracted.
Example3x and 7x are like terms → 3x + 7x = 10x
3x and 3x² are NOT like (different power)
3x and 3x² are NOT like (different power)
Expression
A combination of terms joined by + or − signs. It has no equals sign.
Example3x + 5y − 7 is an expression with three terms.
Simplify: 5x + 3y − 2x + 4y − 1
1Group like terms: (5x − 2x) + (3y + 4y) − 1
2= 3x + 7y − 1
3x + 7y − 1
⚠ Never combine unlike terms: 3x + 5 ≠ 8x. The number 5 and the term 3x are unlike — they must stay separate.
B. Algebraic Identities
(a + b)² = a² + 2ab + b²
Square of a sum. Three terms: first squared, twice the product, last squared.
Example(x + 3)² = x² + 2(x)(3) + 3² = x² + 6x + 9
(a − b)² = a² − 2ab + b²
Square of a difference. Same as above but the middle term is negative.
Example(x − 4)² = x² − 2(x)(4) + 4² = x² − 8x + 16
(a + b)(a − b) = a² − b²
Difference of squares. The middle terms cancel out leaving only two terms.
Example(x + 5)(x − 5) = x² − 25
Why (a+b)² = a² + 2ab + b² — The Square Proof
Example 1 — Expand (2x + 3)²
1Use (a+b)² = a² + 2ab + b²
2a = 2x b = 3
3a² = (2x)² = 4x²
42ab = 2 × 2x × 3 = 12x
5b² = 3² = 9
(2x + 3)² = 4x² + 12x + 9
Example 2 — Evaluate 103 × 97 using identity
1Write as (100 + 3)(100 − 3) → matches (a+b)(a−b)
2= 100² − 3² = 10000 − 9
103 × 97 = 9991 (much faster!)
⚠ Most common mistake: (a+b)² ≠ a² + b². Students forget the middle term 2ab. Always write all three terms.
C. Factoring
Method 1 — Take Out Common Factor (GCF)
Find the biggest number/variable that divides all terms. Put it outside brackets.
Example6x² + 9x → GCF is 3x → 3x(2x + 3)
Method 2 — Difference of Squares
When you have a² − b² (subtraction, both perfect squares). Factors into (a+b)(a−b).
Examplex² − 16 = x² − 4² = (x + 4)(x − 4)
Method 3 — Split the Middle Term
For x² + bx + c: find two numbers whose product = c and sum = b. Those become the bracket values.
Examplex² + 5x + 6 → need product=6, sum=5 → 2 and 3 → (x+2)(x+3)
Example 1 — Factor x² + 7x + 12
1Need two numbers: product = 12 and sum = 7
2Try: 3 × 4 = 12 ✓ 3 + 4 = 7 ✓
3Answer: (x + 3)(x + 4)
✓Check: (x+3)(x+4) = x²+4x+3x+12 = x²+7x+12 ✓
x² + 7x + 12 = (x + 3)(x + 4)
Example 2 — Factor x² − 25
1Recognise: x² − 25 = x² − 5² (difference of squares)
2= (x + 5)(x − 5)
x² − 25 = (x + 5)(x − 5)
Always try GCF first before any other method. If all terms share a factor, take it out first.
Quick MCQ Revision
| Identity | Formula |
|---|---|
(a+b)² | a² + 2ab + b² — never forget 2ab! |
(a−b)² | a² − 2ab + b² |
a²−b² | (a+b)(a−b) |
| Factoring order | 1) GCF 2) Difference of squares 3) Split middle |