Polynomials path · A07

Before this lesson: Vieta’s formulas, Algebraic identities and factorisation

Your goal: Rewrite symmetric expressions in a smaller set of quantities.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Symmetric polynomials: the key idea

An expression is symmetric if exchanging any two variables leaves it unchanged. For two variables, s=x+y and p=xy often compress the problem: x²+y²=s²−2p and x³+y³=s³−3sp. For three variables use s₁=x+y+z, s₂=xy+yz+zx and s₃=xyz. Power sums follow by expanding these or using the polynomial satisfied by each variable. A cyclic expression need not be symmetric. When solving a symmetric system, reconstruct the variables as roots and check that those roots meet the required domain.

A worked example

Real x,y satisfy x+y=7 and x²+y²=25. Find all ordered pairs.

The identity 25=49−2xy gives xy=12. Therefore x,y are roots of t²−7t+12=(t−3)(t−4). The ordered pairs are (3,4) and (4,3), and both check. Listing only one would miss the symmetry.

Your turn: change one thing

If x+y=4 and xy=−1, find x⁴+y⁴.

Try this on paper before opening the explanation.

Compare your reasoning

First x²+y²=16+2=18. Then x⁴+y⁴=(x²+y²)²−2x²y²=324−2=322.

Pause and check

A trap to avoid: Treating a cyclic expression as symmetric or losing possible triples.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove that x³+y³+z³−3xyz=(x+y+z)(x²+y²+z²−xy−yz−zx).

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Expand the right side. The mixed terms x²y,x²z,xy²,xz²,y²z,yz² cancel in pairs, leaving x³+y³+z³−3xyz. Alternatively write x³+y³+z³=s₁³−3s₁s₂+3s₃ and subtract 3s₃. In particular, when x+y+z=0 the sum of cubes equals 3xyz.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. Which expression is symmetric in x and y?

Foundation

  1. 2x+y
  2. x/y for all nonzero inputs
  3. x²+y²
  4. x²−y²
Hint

Swap the variable names.

Answer and reasoning

x²+y². x²+y² is unchanged; the other displayed expressions generally change.

2. If x+y=6 and xy=5, find x²+y².

Foundation

  1. 36
  2. 26
  3. 16
  4. 31
Hint

Use s²−2p.

Answer and reasoning

26. 36−10=26.

3. If x+y=4 and xy=3, find x³+y³.

Core

  1. 16
  2. 40
  3. 64
  4. 28
Hint

Use s³−3sp.

Answer and reasoning

28. 64−3×4×3=28.

4. x+y=5 and xy=6. What are all ordered pairs?

Core

  1. (−2,−3) and (−3,−2)
  2. (2,3) and (3,2)
  3. Only (2,3)
  4. (1,4) and (4,1)
Hint

Reconstruct a quadratic with roots x,y.

Answer and reasoning

(2,3) and (3,2). t²−5t+6=(t−2)(t−3), giving both orderings.

5. For three variables, does cyclic symmetry always imply full symmetry?

Stretch

  1. No
  2. Yes
  3. Only for integers
  4. Only for positive variables
Hint

Compare cycling with swapping two variables.

Answer and reasoning

No. Cycling is a smaller set of permutations. For example x²y+y²z+z²x can change when x and y are exchanged.

6. If x+y+z=0 and xyz=4, find x³+y³+z³.

Stretch

  1. 0
  2. 4
  3. 16
  4. 12
Hint

Use the sum-of-cubes identity.

Answer and reasoning

12. The factor x+y+z vanishes, so the sum of cubes is 3xyz=12.

7. A symmetric expression is unchanged under what?

Foundation

  1. Every permutation of its variables
  2. Only one chosen substitution
  3. Changing every sign always
  4. Increasing one variable
Hint

Use the definition of full symmetry.

Answer and reasoning

Every permutation of its variables. Relabelling variables in any order leaves the expression unchanged.

8. If a+b=7 and ab=10, what is a³+b³?

Core

  1. 133
  2. 343
  3. 203
  4. 70
Hint

Use (a+b)³−3ab(a+b).

Answer and reasoning

133. 343−3·10·7=133.

9. Is ab²+bc²+ca² always symmetric in a,b,c?

Stretch

  1. Only if the letters are sorted
  2. It is a constant
  3. No, it is cyclic but not fully symmetric
  4. Yes, every cyclic expression is symmetric
Hint

Swap a and b in a test case.

Answer and reasoning

No, it is cyclic but not fully symmetric. At (1,2,3) it equals 25, while swapping a,b gives 23.

Choose your next step

Continue to Common roots of polynomials. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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