Factors of Polynomials (Copy)
FACTORS OF POLYNOMIALS — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Polynomial | Expression with powers of x | Thinking fractions are polynomials |
| Degree of Polynomial | Highest power of x | Adding powers together |
| Linear Polynomial | ax + b | Confusing with equation |
| Quadratic Polynomial | ax² + bx + c | Forgetting a ≠0 |
| Cubic Polynomial | ax³ + bx² + cx + d | Ignoring highest power |
| Common Factor | ax + ay = a(x + y) | Leaving common factor inside |
| Difference of Squares | a² − b² = (a − b)(a + b) | Writing (a − b)² |
| Perfect Square | a² + 2ab + b² = (a + b)² | Missing middle term |
| Perfect Square | a² − 2ab + b² = (a − b)² | Wrong sign |
| Quadratic Factorisation | x² + bx + c = (x + p)(x + q) | Wrong number pair |
| Product-Sum Rule | pq = c and p + q = b | Mixing addition and multiplication |
| Factorising with Leading Coefficient | Split middle term | Wrong split values |
| Factor Theorem | f(a) = 0 → (x − a) factor | Forgetting sign in factor |
| Remainder Theorem | f(a) gives remainder | Mixing with factor theorem |
| Repeated Factor | (x − a)² | Writing different roots |
| Root-Factor Relationship | x = a → factor (x − a) | Writing (x + a) |
| Polynomial Equation | Set equal to 0 first | Solving without rearranging |
| Zero Product Rule | ab = 0 → a = 0 or b = 0 | Setting both together |
| Cubic Factorisation | Find one factor first | Random factor guessing |
| Synthetic Division | Fast polynomial division | Using wrong coefficient order |
| Long Division | Divide highest powers first | Dividing constants first |
| Algebraic Fraction Simplification | Factorise completely before cancelling | Cancelling unfactorised terms |
| Restrictions | Denominator ≠0 | Forgetting restrictions |
| Expansion Identity | (a + b)² = a² + 2ab + b² | Writing a² + b² |
| Expansion Identity | (a − b)² = a² − 2ab + b² | Wrong sign in middle |
| Cubic Expansion | (a + b)³ = a³ + 3a²b + 3ab² + b³ | Missing terms |
| Quartic Polynomial | Degree 4 polynomial | Calling cubic |
| Polynomial Roots | x-values where polynomial = 0 | Giving y-values |
| x-intercepts | Roots of polynomial | Mixing with y-intercept |
| Graph Relationship | Root x = a → graph crosses at x = a | Wrong coordinate |
| Consecutive Factors | x(x + 1)(x − 1) | Expanding incorrectly |
| Repeated Root Graph | Graph touches x-axis | Thinking graph crosses |
| Distinct Roots | Graph crosses x-axis separately | Drawing tangent |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK FACTORISATION IDENTITIES
| Identity | Formula |
|---|---|
| Difference of Squares | a² − b² = (a − b)(a + b) |
| Perfect Square | a² + 2ab + b² = (a + b)² |
| Perfect Square | a² − 2ab + b² = (a − b)² |
| Cubic Identity | (a + b)³ = a³ + 3a²b + 3ab² + b³ |
| Cubic Identity | (a − b)³ = a³ − 3a²b + 3ab² − b³ |
QUICK FACTOR THEOREM TABLE
| If | Then |
|---|---|
| f(a) = 0 | (x − a) is a factor |
| f(a) ≠0 | (x − a) is not a factor |
| f(a) = remainder | Remainder theorem |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
COMMON FACTORISATIONS
| Expression | Factorised Form |
|---|---|
| x² − 9 | (x − 3)(x + 3) |
| x² + 5x + 6 | (x + 2)(x + 3) |
| x² − 5x + 6 | (x − 2)(x − 3) |
| x² − 4x + 4 | (x − 2)² |
| x² + 6x + 9 | (x + 3)² |
| 4x² − 25 | (2x − 5)(2x + 5) |
| x³ − x | x(x − 1)(x + 1) |
| x³ − 8 | (x − 2)(x² + 2x + 4) |
| x³ + 8 | (x + 2)(x² − 2x + 4) |
QUICK GRAPH CONNECTIONS
| Graph Behaviour | Meaning |
|---|---|
| Crosses x-axis at x = a | (x − a) factor |
| Touches x-axis at x = a | Repeated root |
| 2 roots | Quadratic crosses twice |
| 3 roots | Cubic crosses three times maximum |
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Factorising quadratics | Find product and sum pair |
| Cubics | Test simple roots first |
| Difference of squares | Look for subtraction |
| Perfect squares | Check middle term |
| Algebraic fractions | Factorise fully first |
| Factor theorem | Substitute candidate root |
| Repeated roots | Look for squared factors |
| Graph questions | Roots = x-intercepts |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Wrong signs in factors | Entire factorisation incorrect |
| Forgetting common factor | Incomplete simplification |
| Wrong factor pair | Wrong roots |
| Missing restrictions | Invalid algebraic fraction |
| Cancelling unfactorised terms | Illegal simplification |
| Wrong factor theorem sign | Incorrect factor |
| Incomplete expansion | Algebra error |
| Missing repeated root | Wrong graph interpretation |
| Not setting equation to 0 | Cannot solve correctly |
| Poor graph-root connection | Wrong intercepts |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.