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This scheme of work for A-Level algebraic methods builds on earlier work at GCSE with factorising quadratics and writing a short algebraic argument. It develops the skill of dividing a cubic and constructing a proof, and these ideas are used throughout the AS Pure course.
Students learn to divide a polynomial by a linear factor, then use the factor theorem to find a root. They write a proof by deduction, then choose exhaustion or a counter-example.
By the end of this unit students will be able to:
Students should be secure with the following before beginning this unit.
Divide by (x minus a). The quotient is a quadratic. A remainder of 0 means (x minus a) is a factor. Using quadratic factorising on a cubic, or stopping with a remainder still sitting there, leaves the cubic unsolved.
Try small integers. Once a factor is found, divide, then factorise the quadratic. Substituting minus a when the factor is (x plus a), or mishandling the cubic term, gives a non-zero that is not the true remainder.
Write n as an integer, or 2k, or 2k plus 1, then expand. The last line must say what has been proved. An expansion with no conclusion is algebra, not a proof.
If the statement is about odd and even, check both. If it is sometimes true, show a case that works and a case that fails. One worked example does not prove a general claim. One failed trial, with no written statement that the claim is false, is not a disproof.
| Misconception | Teaching focus |
|---|---|
| Using long division on a quadratic that factorises in one line. | Factorise first. Division is for a cubic, or when a linear factor is already known. |
| Substituting a negative into the cubic term and dropping the minus. | Write each term on its own line. Cube the negative first, then multiply by the coefficient. |
| Testing f(a) when the intended factor is (x plus a), so the wrong value is used. | (x plus a) is a factor when f(minus a) is 0. Write the value you are substituting before you compute. |
| Stopping a proof after the expansion, with no written conclusion. | The last sentence must state what has been shown, in words. |
| Offering one example as a general proof, or one failed attempt as a counter-example without saying the claim is false. | A proof needs every case, or an identity. A disproof needs one clear counter-example and a sentence that the statement is false. |
Every lesson in this unit is already built. A Mr Mathematics membership gives you the presentation, worksheet, question generator and video tutorial for all four lessons, alongside the full Key Stage 3, GCSE, IGCSE and A-Level libraries.
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Students learn how to divide polynomials using algebraic long division and factorisation. They start by rewriting cubic expressions in the form (x ± p)(ax^2 + bx + c), linking division to factor structure.
As learning progresses, they find remainders, solve polynomial equations, and fully factorise functions.
Students learn how to calculate the roots of a cubic expression using the factor theorem.
Later, as learning progresses, they use the factor theorem to work out the coefficients of x3, x2 and x.
Students learn how to derive a mathematical proof by deduction.
Later, as learning progresses, they derive geometrical properties using trigonometry and other known facts.
Students learn about the different methods of proof.
Later, as learning progresses, they consider proofs by exhaustion, deduction and how to disprove a conjecture using counter-examples.
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