Algebraic Identities — List, Proofs, and Examples
Book A Free Math Class
Algebraic Identities — List, Proofs, and Examples
TL;DR
Algebraic identities are equations that stay true for every value of their variables — for example (a+b)²=a²+2ab+b² and a²−b²=(a+b)(a−b). This article gives you the full standard list (square, cube, and three-variable identities), a geometric and an algebraic proof, six worked examples, and the slips that cost the most marks.
Algebraic Identities — Definition, List, Proofs, and Examples
Algebraic identities are equations that stay true for every value of their variables — for example (a+b)²=a²+2ab+b² and a²−b²=(a+b)(a−b). This article gives you the full standard list (square, cube, and three-variable identities), a geometric and an algebraic proof, six worked examples, and the slips that cost the most marks.
The equation that is true no matter what you put into it
Most equations are fussy. 2x+3=11 is true only when x=4 — change the value and it breaks. But (a+b)²=a²+2ab+b² never breaks. Pick any numbers you like for a and b, and both sides land on the same answer. That unconditional truth is what makes an identity a tool you can lean your whole weight on.
What is an Algebraic Identity?
An algebraic identity is an equation in which the left-hand side equals the right-hand side for all values of the variables involved. Substitute anything, and equality holds.
Contrast that with an ordinary equation, which is only true for particular values. x+2=5 is true at x=3 and false everywhere else. (a+b)²=a²+2ab+b² is true everywhere. That "everywhere" is the defining feature.
The standard algebraic identities list
Here is the working set, grouped by degree. These are the ones worth knowing cold.
Two-variable square identities
| Identity | Use |
|---|---|
| (a+b)²=a²+2ab+b² | Expand a squared sum |
| (a−b)²=a²−2ab+b² | Expand a squared difference |
| a²−b²=(a+b)(a−b) | Factor a difference of squares |
| (x+a)(x+b)=x²+(a+b)x+ab | Expand a product of two linear binomials |
Cube identities
| Identity | Use |
|---|---|
| (a+b)³=a³+3a²b+3ab²+b³ | Expand a cubed sum |
| (a−b)³=a³−3a²b+3ab²−b³ | Expand a cubed difference |
| a³+b³=(a+b)(a²−ab+b²) | Factor a sum of cubes |
| a³−b³=(a−b)(a²+ab+b²) | Factor a difference of cubes |
Three-variable identity
| Identity | Use |
|---|---|
| (a+b+c)²=a²+b²+c²+2ab+2bc+2ca | Expand a squared trinomial |
| a³+b³+c³−3abc=(a+b+c)(a²+b²+c²−ab−bc−ca) | A useful factoring identity |
How do you prove an Algebraic Identity?
Two routes prove an identity: a geometric proof (areas or volumes) and an algebraic proof (expanding and collecting terms). Both confirm the two sides are genuinely the same expression.
Examples of Algebraic Identities
Six problems, easier to harder, each one matching the expression to the right identity before substituting.
Example 1
Expand (x+7)².
Use (a+b)²=a²+2ab+b² with a=x, b=7.
Final answer: x²+14x+49.
Example 2
Expand (3y−5)².
Use (a−b)²=a²−2ab+b².
Final answer: 9y²−30y+25.
Example 3
Factor x²−81. Recognise a difference of squares: 81=9².
Final answer: (x+9)(x−9).
Example 4
Use (x+a)(x+b)=x²+(a+b)x+ab to expand (x+4)(x+6).
Final answer: x²+10x+24.
Example 5
Evaluate 103² using an identity. Use (a+b)²=a²+2ab+b²: 103²=100²+2(100)(3)+3².
Final answer: 10609.
Example 6
Factor 27x³+8 using the sum-of-cubes identity.
Final answer: (3x+2)(9x²−6x+4).
Common Mistakes With Algebraic Identities
Mistake 1: Forgetting the middle term
Where it slips in: Any time a binomial is squared.
Don't do this: (a+b)²=a²+b².
The correct way: (a+b)²=a²+2ab+b².
Mistake 2: Confusing (a−b)² with a²−b²
Where it slips in: A squared difference and a difference of squares look similar.
Don't do this: Treat (x−3)² as x²−9.
The correct way: (x−3)²=x²−6x+9.
Mistake 3: Sign errors in the cube formulas
Where it slips in: a³−b³ versus a³+b³.
Don't do this: Write a³−b³=(a−b)(a²−ab+b²).
The correct way: a³−b³=(a−b)(a²+ab+b²).
Conclusion
Algebraic identities are equations true for every value of their variables — that unconditional truth is what separates them from ordinary equations.
The core list is small: three square identities, four cube identities, and a couple of three-variable ones.
Every identity can be proved geometrically (areas) or algebraically (expand and collect).
The most expensive mistakes are dropping the middle term and confusing a squared difference with a difference of squares.
Master these once and they pay off across factoring, mental arithmetic, and calculus for years.
Practice These to Solidify Your Understanding
- Expand (2m+5)².
- Factor 49x²−16.
- Factor x³−125.