Further algebra and series

What further algebra and series covers

What the roots of a polynomial say about its coefficients without ever being found, sums that collapse like telescopes, and functions rebuilt as infinite polynomials. Core Pure content, examinable on both compulsory papers of 9FM0. The first lesson leans on the conjugate root pairs from Complex numbers, and the last leans on the standard derivatives from A level differentiation.

The main ideas

• Sums and products of roots read straight from the coefficients of a quadratic, cubic or quartic, with the signs alternating.
• Symmetric functions such as the sum of the squares of the roots, evaluated without solving anything.
• New polynomials whose roots are a transformation of the old ones, built by a stated substitution or from the new sum and product.
• Cubics and quartics with real coefficients solved from one given complex root, using the conjugate pair.
• The standard results for the sums of r, r squared and r cubed, combined and factorised to sum a polynomial series.
• The method of differences, where each term splits into a difference and the sum telescopes to its ends.
• Maclaurin series by repeated differentiation at zero, the standard series, and their intervals of validity.

The results it turns on

for ax² + bx + c: the sum of the roots is −b/a and their product is c/a

the root relations, with the same alternating pattern for cubics and quartics

α² + β² + γ² = (Σα)² − 2Σαβ

the identity a symmetric function question expects

Σr = ½n(n + 1), with the Σr² and Σr³ results in the booklet

the standard results a polynomial series is assembled from

Σ(f(r) − f(r + 1)) leaves only the terms at the two ends

the method of differences

f(x) = f(0) + xf'(0) + x²f''(0)/2! + …

the Maclaurin series of a function

ln(1 + x) holds for −1 < x ≤ 1, and arctan x for −1 ≤ x ≤ 1

the two standard series whose windows are limited

Where it usually goes wrong

• Divide through by the leading coefficient before quoting any root relation. Skipping that step corrupts every relation that follows.
• The standard summation results start at r = 1, so a sum starting at r = k means subtracting the sum up to k - 1.
• A gap of two in the denominators of a telescoping sum leaves two survivors at each end rather than one, so write at least two rows at both ends before cancelling.
• After substituting into the series for ln(1 + x) or arctan x, the validity window has to be transformed by the same substitution.

Where to start

Roots of polynomials first, and cover complex conjugate pairs before it. Summing series and the method of differences second, since the standard results are quick marks and the telescope is a separate technique. Maclaurin series last, because it calls on the whole differentiation shelf.