Summary
Factors of polynomials involve expanding and factorising expressions using algebraic methods. Expansion involves multiplying terms, while factorisation is the reverse process.
- Expansion — multiplying each term outside the bracket with each term inside the bracket. Example: 2x(3x + y - 4z) = 6x² + 2xy - 8xz
- Factorisation — taking out the common factor from terms. Example: x² + x = x(x+1)
- Factorising Quadratics — finding two numbers that multiply to the constant term and add to the linear coefficient. Example: x² + 11x + 24 = (x+3)(x+8)
- Division of Polynomials — similar to long division, where the degree of the quotient is the degree of the dividend minus the degree of the divisor. Example: The degree of the remainder is at most one less than the degree of the divisor.
- Factor Theorem — if f(a) = 0, then (x-a) is a factor of f(x). Example: If f(x) = x³ - 1, then (x-1) is a factor.
- Remainder Theorem — the remainder of f(x) divided by x-a is f(a). Example: The remainder when f(x) is divided by x-1 is f(1).
Exam Tips
Key Definitions to Remember
- Expansion: Multiplying terms inside and outside brackets.
- Factorisation: Taking out common factors from terms.
- Factor Theorem: If f(a) = 0, then (x-a) is a factor.
- Remainder Theorem: The remainder of f(x) divided by x-a is f(a).
Common Confusions
- Mixing up expansion and factorisation.
- Forgetting to check both multiplication and addition conditions in quadratic factorisation.
Typical Exam Questions
- How do you expand 2x(3x + y - 4z)? 6x² + 2xy - 8xz
- Factorise x² + 11x + 24. (x+3)(x+8)
- What is the remainder when f(x) is divided by x-1? f(1)
What Examiners Usually Test
- Ability to expand and simplify expressions.
- Correct application of factorisation techniques.
- Understanding and applying the factor and remainder theorems.