The relationships between the roots and coefficients of quadratic, cubic and quartic equations, symmetric functions of the roots, and forming a new equation whose roots are a given function of the original roots.

What this dot point is asking

OCR wants you to use the relationships between the roots and coefficients of quadratic, cubic and quartic equations, to evaluate symmetric functions of the roots (such as $\sum\alpha^2$ or $\sum\dfrac{1}{\alpha}$) using identities, and to form a new polynomial whose roots are a given function of the original roots, either through the new sums and products or by a substitution.

Roots and coefficients

The elementary symmetric functions of the roots are read directly off the coefficients, with signs that alternate. These are the starting point for every roots question.

Symmetric functions via identities

Functions of the roots that are symmetric (unchanged by permuting the roots) can be written in terms of the elementary symmetric functions, and hence in terms of the coefficients, without ever finding the roots. The key identities convert sums of powers or reciprocals.

Forming a new equation: new sums and products

To build an equation whose roots are a function of the originals, compute the new sum and product (or, for a cubic, the three elementary symmetric functions) and assemble the polynomial. For a quadratic, the new equation is $x^2 - (\text{new sum})x + (\text{new product}) = 0$.

Forming a new equation: substitution

Often quicker is to substitute. If the new roots are $y = g(\alpha)$, write $\alpha = g^{-1}(y)$ and substitute into the original equation, then tidy. For new roots $\alpha + 2$, substitute $x \to x - 2$; for new roots $2\alpha$, substitute $x \to \dfrac{x}{2}$; for reciprocals $\dfrac{1}{\alpha}$, substitute $x \to \dfrac{1}{x}$ and clear denominators. The substitution method is especially efficient for a cubic or quartic, where computing all the new elementary symmetric functions by hand is tedious; a single substitution and a tidy-up gives the whole new equation at once. The two methods are equivalent and either earns full marks, so choose the substitution route when the transformation is a simple shift, scaling or reciprocal, and the new-sums-and-products route when the new roots are a less standard combination (such as $\alpha\beta$, $\beta\gamma$, $\gamma\alpha$ for a cubic) that does not correspond to a clean substitution. Whichever you use, present the final equation with integer coefficients unless told otherwise, multiplying through to clear any fractions.

Roots and coefficients link the series strand to complex numbers (a real polynomial's complex roots come in conjugate pairs) and reward careful, symmetric algebra.

Try this

Q1. The quadratic $x^2 + 7x + 12 = 0$ has roots $\alpha, \beta$. State $\alpha + \beta$ and $\alpha\beta$. [2 marks]

• Cue. $\alpha + \beta = -7$, $\alpha\beta = 12$.

Q2. For a cubic with $\sum\alpha = 4$ and $\sum\alpha\beta = 1$, find $\sum\alpha^2$. [2 marks]

• Cue. $\sum\alpha^2 = (\sum\alpha)^2 - 2\sum\alpha\beta = 16 - 2 = 14$.