The coefficients of a polynomial already contain the sums and products of its roots. That means you can answer questions about the roots without ever finding them, which is the whole point of the topic.
Syllabus points covered
- Recall and use the relations between the roots and coefficients of quadratic, cubic and quartic equations
- Use a substitution to find the equation whose roots are related to those of a given equation
- Evaluate symmetric functions of the roots, including sums of squares and cubes
- Use the relations to solve problems involving unknown coefficients
Common mistakes
- Sign errors in the relations — for a cubic the sum of roots is −b/a but the sum of products in pairs is +c/a.
- Trying to find the roots explicitly when the symmetric functions are all that is needed.
- Forgetting that Σα² = (Σα)² − 2Σαβ rather than simply (Σα)².
- Substituting for the new variable the wrong way round when transforming an equation.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Practice worksheet
- Worksheet answers
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questions
- Mark scheme
- Step-by-step worked solutionsComing soon
Looking for whole papers by session? All Further Mathematics past papers