Complex Algebraic Manipulation
Complex algebraic manipulation involves factorising, expanding, and simplifying advanced expressions including quadratics, cubics, and rational expressions.
What You Need to Know
Key Concept Diagram
Factorising by grouping applies when a four-term polynomial can be split into two pairs
The difference of two squares: a^2 - b^2 = (a+b)(a-b)
Sum and difference of cubes: a^3 + b^3 = (a+b)(a^2 - ab + b^2)
Rational expressions are simplified by factorising numerator and denominator then cancelling common factors
Long division of polynomials divides a higher-degree polynomial by a lower-degree one
Key Vocabulary
Factorisation
Expressing an expression as a product of its factors
Rational expression
A fraction where numerator and/or denominator are polynomials
Polynomial long division
A method to divide polynomials similar to numerical long division
Completing the square
Rewriting a quadratic in the form (x + p)^2 + q
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Factorise x^2 - 9.
Question 2
Simplify (x^2 - 4) / (x + 2).
Question 3
Which expression is equivalent to x^2 + 6x + 9?
Key Concepts Summary
- ●Factorising by grouping applies when a four-term polynomial can be split into two pairs
- ●The difference of two squares: a^2 - b^2 = (a+b)(a-b)
- ●Sum and difference of cubes: a^3 + b^3 = (a+b)(a^2 - ab + b^2)
- ●Rational expressions are simplified by factorising numerator and denominator then cancelling common factors
- ●Long division of polynomials divides a higher-degree polynomial by a lower-degree one