Edexcel IAL - Pure Maths 1- 1.10 Algebraic Manipulation of Polynomials- Study notes - New syllabus
Edexcel IAL – Pure Maths 1- 1.10 Algebraic Manipulation of Polynomials -Study notes- New syllabus
Edexcel IAL – Pure Maths 1- 1.10 Algebraic Manipulation of Polynomials -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 1.10 Algebraic Manipulation of Polynomials
Algebraic Manipulation of Polynomials
Polynomials are algebraic expressions involving powers of \( x \) with non-negative integer exponents. Students must be familiar with expanding brackets, collecting like terms, and factorising polynomials up to degree 3.
A polynomial may be written using function notation, for example:
\( f(x) = x^3 + 4x^2 + 3x \)
Key Skills in Polynomial Manipulation
| Skill | Description / Method |
| Expanding brackets | Use distributive law: \( a(b + c) = ab + ac \) and \( (x + a)(x + b) = x^2 + (a + b)x + ab \) |
| Collecting like terms | Combine terms with the same power of \( x \). Example: \( 4x^2 – 3x + 7x^2 + 5x = 11x^2 + 2x \) |
| Factorising quadratics | Factorise \( ax^2 + bx + c \) using: • common factor • splitting the middle term • use of quadratic identities |
| Factorising cubic expressions | For \( x^3 + px^2 + qx + r \): • Take out common factors • Use grouping, e.g. \( x^3 + 4x^2 + 3x = x(x^2 + 4x + 3) \) • Factorise the remaining quadratic |
Factorisation Strategies
- Always first look for a common factor.
- Quadratics may factorise into two linear brackets.
- Cubic polynomials may factorise by grouping or by taking out \( x \) if present.
- Check by expanding to confirm correctness.
Example
Expand and simplify:
\( (x + 3)(x + 5) \)
▶️ Answer / Explanation
Expand:
\( x(x + 5) + 3(x + 5) \)
\( = x^2 + 5x + 3x + 15 \)
\( = x^2 + 8x + 15 \)
Example
Factorise the quadratic:
\( x^2 + 7x + 10 \)
▶️ Answer / Explanation
Find two numbers with sum 7 and product 10: \( 5 \) and \( 2 \)
So:
\( x^2 + 7x + 10 = (x + 5)(x + 2) \)
Example
Factorise the polynomial using grouping:
\( x^3 + 4x^2 + 3x \)
▶️ Answer / Explanation
Common factor:
\( x^3 + 4x^2 + 3x = x(x^2 + 4x + 3) \)
Factorise the quadratic:
\( x^2 + 4x + 3 = (x + 3)(x + 1) \)
Final factorisation:
\( x(x + 3)(x + 1) \)