If α and β are the zeroes of x² − 4x + 3, find a quadratic polynomial whose zeroes are 1/α and 1/β.

1. 1 · Write what is given and what is to be found. α and β are zeroes of x² − 4x + 3. To find: a polynomial with zeroes 1/α and 1/β.
2. 2 · Write the formula. α + β = −b/a = 4 and αβ = c/a = 3. Required polynomial: k[x² − (sum)x + product].
3. 3 · Find the new sum. 1/α + 1/β = (α + β)/(αβ) = 4/3
4. 4 · Find the new product. (1/α)(1/β) = 1/(αβ) = 1/3
5. 5 · Write the polynomial. x² − (4/3)x + 1/3. Take k = 3: 3x² − 4x + 1.
6. 6 · Check and box the answer. 3x² − 4x + 1 = (3x − 1)(x − 1), with zeroes 1/3 and 1. The old zeroes are 3 and 1, whose reciprocals are 1/3 and 1. Therefore the answer is 3x² − 4x + 1.

Scoring tip: Never find the old zeroes first unless asked. Using α + β and αβ is faster and is what the marking scheme expects.