Simplifying fractions with unfactorised quadratic in denominator or numerator

Task goals

• Simplify a fraction with a quadratic numerator by factorising out the common variable factor and cancelling it against a monomial denominator.
• Recognise a difference of two squares in the numerator, factorise it into two brackets, and cancel the bracket that matches the linear denominator.
• Factorise a monic quadratic trinomial numerator into two brackets and cancel the bracket that matches the linear denominator.
• State the restriction on the variable that keeps the denominator non-zero, and name the factor that cancels, when simplifying such a fraction.
• Simplify a fraction that has a quadratic in both the numerator and the denominator by factorising each one fully, including removing any numeric common factor, before cancelling the shared factor.
• Factorise non-monic quadratics, where the coefficient of the squared term is greater than one, in both the numerator and the denominator and cancel the common bracket.
• Multiply, divide, add, or subtract algebraic fractions whose numerators or denominators must first be factorised from unfactorised quadratics, and state any restrictions on the variable.
• Explain why cancelling individual terms instead of whole factors is incorrect, and construct an algebraic fraction that simplifies to a given result.