Roots of polynomial equations
Connect polynomial coefficients with symmetric expressions in the roots, then transform roots using substitutions.
Exam scope and assessment notes
Mapped to Cambridge International AS & A Level Further Mathematics 9231. These original explanations and self-checks are independent study support, not Cambridge questions, marks or grades. The syllabus assumes prior knowledge of the relevant 9709 Mathematics content; check each component's prerequisites with your teacher. Full-paper assessment and human subject-specialist review remain outstanding.
Before you begin
- Cambridge expects learners to know the whole Cambridge International A Level Mathematics (9709) content. Paper 1 further pure mathematics draws on 9709 Papers 1 and 3. Confirm your exam year and component route with your school.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What makes a Further Mathematics solution convincing?
What you will learn
- Use relations between roots and coefficients for equations of degree 2, 3 and 4
- Find symmetric expressions and unknown coefficients
- Use substitutions when new roots are simple functions of the original roots
Coefficients store symmetric information
For a monic cubic with roots α, β and γ, α + β + γ is the negative coefficient of x²; αβ + αγ + βγ is the coefficient of x; and αβγ is the negative constant term.
For a monic quartic, the alternating signs continue: the sum of roots is the negative x³ coefficient, the sum of pairwise products is the x² coefficient, the sum of triple products is the negative x coefficient, and the product is the constant term.
These relations let you evaluate expressions such as α² + β² + γ² without solving for each root: square the sum and subtract twice the pairwise sum.
A substitution transfers roots to a new equation
If the original root is α and the new root is u = α + 1, replace α by u − 1 in the original equation. Clear denominators and simplify to obtain an equation for u.
When roots become reciprocals, a polynomial of degree n can be reversed by replacing powers in descending order. Check that zero is not an original root before using reciprocals.
Only transform the roots the question specifies. A transformed equation should retain the correct degree and include every transformed root, with multiplicity.
Pause and explain
Which habit best checks your work on roots of polynomial equations?
Worked example
A monic cubic has roots α, β and γ and is x³ − 4x² + x + 6 = 0. Find α² + β² + γ².
Show the worked solution
- The sum of roots is α + β + γ = 4.
- The pairwise sum is αβ + αγ + βγ = 1.
- Square the root sum: α² + β² + γ² = (α + β + γ)² − 2(αβ + αγ + βγ) = 16 − 2 = 14.
Answer 14
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For x³ − 6x² + 7x + 2 = 0 with roots α, β, γ, find α² + β² + γ².
Give me a hint
Read the sum and pairwise sum from the coefficients, then use (α+β+γ)² − 2(αβ+αγ+βγ).
Compare my reasoning
- The root sum is 6 and the pairwise sum is 7.
- The sum of squares is 6² − 2(7).
- Calculate 36 − 14.
22
Look for these in your work
- Both coefficient relations used
- Identity applied correctly
- Exact value stated
A monic cubic has roots α, β, γ. Its root sum is 3 and pairwise product sum is −4. Find the coefficient of x² and the coefficient of x.
Give me a hint
For a monic cubic, the x² coefficient is minus the root sum; the x coefficient is the pairwise sum.
Compare my reasoning
- The x² coefficient is −3.
- The x coefficient is −4.
- Keep the constant term separate; it depends on the product of all three roots.
x² coefficient −3; x coefficient −4.
Look for these in your work
- Alternating sign for the x² coefficient
- Pairwise sum placed at x
- Product not confused with pairwise sum
The roots of a cubic are α, β, γ. A new equation is required with roots α+2, β+2 and γ+2. Describe how to obtain it from the original polynomial p(x)=0.
Give me a hint
If u=α+2, then the old root is α=u−2.
Compare my reasoning
- Write p(u−2)=0.
- Expand and collect powers of u; the leading coefficient remains unchanged for a monic cubic.
- The resulting polynomial has the three shifted roots, including multiplicity.
Substitute x=u−2 into p(x), expand, and write the resulting equation in u.
Look for these in your work
- Substitution direction correct
- New variable connected to the old root
- Multiplicity preserved
Common mistakes
- Forgetting the alternating signs in the coefficient relations
- Solving a polynomial when only a symmetric expression is needed
- Using reciprocal roots without checking whether zero is a root
What can you explain now?
For x³ + 5x² − 2x + 7 = 0 with roots α, β, γ, what is α + β + γ?
Compare with the explanation
−5
For a monic cubic, the sum of roots is the negative coefficient of x².
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Make it stick
How do the coefficients of a monic cubic encode the sum, pairwise sum and product of its roots?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.