Roots of polynomial equations

Further Maths · Further Pure Mathematics 1. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

You can learn a lot about the roots of a polynomial equation without solving it. The coefficients tell you the sum of the roots, the sum of the products of the roots taken two at a time, and so on, down to the product of the roots. The signs alternate: −, +, −, +.

Any expression that stays the same when the roots are swapped (a symmetric function), such as α² + β² + γ², can be built from these sums and products.

To find a new equation whose roots are related to the old ones, use a substitution: write the new root y in terms of x, make x the subject, and put it into the original equation.

Rules to remember

Quadratic ax² + bx + c = 0: α + β = −b/a and αβ = c/a
Cubic ax³ + bx² + cx + d = 0: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a
Quartic ax⁴ + bx³ + cx² + dx + e = 0: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a
Sum of squares: Σα² = (Σα)² − 2Σαβ
Sum of reciprocals for a cubic: 1/α + 1/β + 1/γ = (αβ + βγ + γα)/(αβγ)
New roots 1/α: put x = 1/y. New roots α²: put x = √y. New roots α + k: put x = y − k. New roots kα: put x = y/k
Each root satisfies the equation, so for x³ + px² + qx + r = 0: Σα³ = −pΣα² − qΣα − 3r

Common mistake

Students forget to divide by a, or get the signs wrong, for example writing α + β + γ = b/a. Divide the whole equation by a first, then read the signs in the order −, +, −, +.

Worked example

The roots of x³ − 2x² + 4x − 3 = 0 are α, β and γ. Find the cubic equation whose roots are α + 1, β + 1 and γ + 1.

Let y = x + 1, so x = y − 1. Substitute: (y − 1)³ − 2(y − 1)² + 4(y − 1) − 3 = 0.
Expand: (y³ − 3y² + 3y − 1) − (2y² − 4y + 2) + (4y − 4) − 3 = 0.
Collect terms: y³ − 5y² + 11y − 10 = 0.
Check: old sum of roots is 2, so the new sum should be 2 + 3 = 5, which matches −(−5)/1 = 5.

Answer: y³ − 5y² + 11y − 10 = 0

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