B: Algebra and functions: Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only).

The idea

Polynomial manipulation underpins much of algebra. Expanding brackets means multiplying every term in one bracket by every term in the other, then collecting like terms; for example (2x + 3)(x² − 4x + 1) gives 2x³ − 8x² + 2x + 3x² − 12x + 3 = 2x³ − 5x² − 10x + 3. Factorisation reverses this, often by extracting a common factor or using known identities. Simple algebraic division by a linear expression can be done by long division or by comparing coefficients. The factor theorem states that (x − a) is a factor of f(x) exactly when f(a) = 0, which allows you to find unknown constants and factorise cubics. Rational expressions are simplified by factorising numerator and denominator fully, then cancelling common factors, remembering that the cancelled factor must not be zero. Division by a linear expression may leave a remainder, which can be written as a fraction.

What earns the marks

• Expands products of polynomials correctly, including brackets with more than two terms.
• Collects like terms accurately and writes the polynomial in descending powers.
• Factorises by taking out common factors and by using the factor theorem to find linear factors.
• Performs algebraic division by a linear expression, showing the quotient and any remainder.
• Uses the factor theorem to test whether a given linear expression is a factor.
• Simplifies rational expressions by factorising and cancelling common factors.
• States any restrictions on the variable where a cancelled factor could be zero.
• Handles division by a linear expression only, as required by the specification.

Common mistakes and how to put them right

• Cancelling terms rather than factors in a rational expression.

  Put it right by factorising first and cancelling whole brackets.

• Sign errors when substituting into the factor theorem, especially with negative values.

  Put it right by writing brackets carefully.

• Forgetting to include a zero coefficient for missing powers during algebraic division.

  Put it right by inserting 0x² or similar placeholders.

Exam tips

• Lay out algebraic division clearly with aligned powers and bring down each term carefully.
• When using the factor theorem, show the substitution and the result f(a) = 0 explicitly.
• After simplifying a rational expression, check whether any value of x must be excluded.