Roots of polynomials and their relationships to coefficients, summation of series using standard results, the method of differences, partial fractions and the Maclaurin series.

What this dot point is asking

AQA wants you to relate the roots of a polynomial to its coefficients, evaluate finite series using the standard results for sums of powers, use the method of differences for telescoping sums, decompose rational functions into partial fractions, and derive and use the Maclaurin series of a function.

Roots and coefficients

For a quadratic with roots $\alpha$ and $\beta$, $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$. For a cubic $ax^3 + bx^2 + cx + d = 0$ with roots $\alpha, \beta, \gamma$, the sum of roots is $-\frac{b}{a}$, the sum of products in pairs $\alpha\beta + \beta\gamma + \gamma\alpha$ is $\frac{c}{a}$, and the product $\alpha\beta\gamma$ is $-\frac{d}{a}$. For a quartic the pattern continues: the symmetric functions alternate in sign as $-\frac{b}{a}, +\frac{c}{a}, -\frac{d}{a}, +\frac{e}{a}$. These are the elementary symmetric functions, and the standard exam tasks build derived quantities from them. The square-of-sum identity $\sum\alpha^2 = (\sum\alpha)^2 - 2\sum\alpha\beta$ and the cube identity $\sum\alpha^3 = (\sum\alpha)^3 - 3(\sum\alpha)(\sum\alpha\beta) + 3\alpha\beta\gamma$ recur often, as does forming a new equation whose roots are a transformation of the originals (such as $\alpha + 1$, $2\alpha$ or $\frac{1}{\alpha}$) by working out the new symmetric functions.

Summing series

The standard results are $\sum_{r=1}^{n} r = \frac{1}{2}n(n+1)$, $\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)$ and $\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$. Split a polynomial summand into these pieces.

The method of differences

If a summand can be written as $f(r) - f(r+1)$, the sum telescopes and almost all terms cancel, leaving only the first and last. Partial fractions are often the way to get this form.

Maclaurin series

The Maclaurin series of $f(x)$ is $f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$, valid where the series converges. Standard expansions for $e^x$, $\sin x$, $\cos x$ and $\ln(1+x)$ are worth knowing.

When a summand is not a simple power of $r$, look for the method of differences instead: a partial fraction split usually turns the term into $f(r) - f(r+1)$ so the sum telescopes, leaving only the boundary terms.