Symmetric Functions and Location of Roots | NIMCET

Quick answer: A symmetric function of the roots α, β is unchanged when α and β are swapped, so it can be written using $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$ . For example, $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ and $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$ . For location of roots of $f(x) = ax^2 + bx + c$ with $a > 0$ : a number k lies between the roots exactly when $f(k) < 0$ .

This builds on quadratic equations: roots and discriminant.

Standard symmetric functions

Let $S = \alpha + \beta$ and $P = \alpha\beta$ .

Expression	In terms of S and P
$\alpha^2 + \beta^2$	$S^2 - 2P$
$(\alpha - \beta)^2$	$S^2 - 4P$
$\alpha^3 + \beta^3$	$S^3 - 3PS$
$\alpha^4 + \beta^4$	$(S^2 - 2P)^2 - 2P^2$
$\frac{1}{\alpha} + \frac{1}{\beta}$	$\frac{S}{P}$
$\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$	$\frac{S^2 - 2P}{P}$
$\alpha^2\beta + \alpha\beta^2$	$PS$

Transforming an equation

If α, β are the roots of $f(x) = 0$ :

New roots	New equation
$\alpha + k$ , $\beta + k$	$f(x - k) = 0$
$k\alpha$ , $k\beta$	$f\left(\frac{x}{k}\right) = 0$
$\frac{1}{\alpha}$ , $\frac{1}{\beta}$	$f\left(\frac{1}{x}\right) = 0$ , that is, $cx^2 + bx + a = 0$
$-\alpha$ , $-\beta$	$f(-x) = 0$

Cubic equations

For $ax^3 + bx^2 + cx + d = 0$ with roots α, β, γ:

Relation	Value
$\alpha + \beta + \gamma$	$-\frac{b}{a}$
$\alpha\beta + \beta\gamma + \gamma\alpha$	$\frac{c}{a}$
$\alpha\beta\gamma$	$-\frac{d}{a}$

Location of roots

Let $f(x) = ax^2 + bx + c$ with $a > 0$ , discriminant D and vertex at $-\frac{b}{2a}$ .

Requirement	Conditions
k lies between the roots	$f(k) < 0$
Both roots greater than k	$D \ge 0$ , $f(k) > 0$ , $-\frac{b}{2a} > k$
Both roots less than k	$D \ge 0$ , $f(k) > 0$ , $-\frac{b}{2a} < k$
Both roots between $k_1$ and $k_2$	$D \ge 0$ , $f(k_1) > 0$ , $f(k_2) > 0$ , $k_1 < -\frac{b}{2a} < k_2$
Exactly one root in $(k_1, k_2)$	$f(k_1)f(k_2) < 0$

For $a < 0$ , apply the same rules to $-f(x)$ , or use $af(k) < 0$ for the first row.

Worked examples

Example 1: α² + β²

If α, β are the roots of $x^2 - 5x + 3 = 0$ , find $\alpha^2 + \beta^2$ and $\alpha^3 + \beta^3$ .

Step 1: $S = 5$ , $P = 3$ .

Step 2: $\alpha^2 + \beta^2 = 25 - 6 = 19$ .

Step 3: $\alpha^3 + \beta^3 = 125 - 45 = 80$ .

Example 2: new equation

Form the equation whose roots are the squares of the roots of $x^2 - 5x + 3 = 0$ .

New sum $= 19$ , new product $= P^2 = 9$ , so $x^2 - 19x + 9 = 0$ .

Example 3: k between the roots

For what m does 2 lie between the roots of $x^2 - mx + 1 = 0$ ?

$f(2) = 4 - 2m + 1 < 0$ , so $m > \frac{5}{2}$ .

Example 4: both roots greater than 1

Find k so that both roots of $x^2 - 2kx + k^2 - 1 = 0$ exceed 1.

Step 1: $D = 4k^2 - 4(k^2 - 1) = 4 > 0$ always.

Step 2: $f(1) = 1 - 2k + k^2 - 1 = k^2 - 2k > 0$ , so $k < 0$ or $k > 2$ .

Step 3: Vertex $k > 1$ .

Answer: $k > 2$ . (Indeed the roots are $k \pm 1$ , and $k - 1 > 1$ .)

Example 5: cubic relation

If α, β, γ are the roots of $x^3 - 6x^2 + 11x - 6 = 0$ , find $\alpha^2 + \beta^2 + \gamma^2$ .

$(\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) = 36 - 22 = 14$ .

Summary table

Idea	Key result
$\alpha^2 + \beta^2$	$S^2 - 2P$
$\alpha^3 + \beta^3$	$S^3 - 3PS$
Reciprocal roots	Swap a and c
k between roots ( $a > 0$ )	$f(k) < 0$
One root in an interval	$f(k_1)f(k_2) < 0$

Exam tips

Rewrite every expression in S and P before substituting.
For location problems, think of the parabola's graph: sign of $f(k)$ , position of the vertex, and $D \ge 0$ .
A quick check: find the roots directly if they are simple, as in Example 4.

Next, see arithmetic progression formulas, or browse the NIMCET Algebra chapter.

Common mistakes

Writing $\alpha^2 + \beta^2 = S^2$ .
Forgetting the vertex condition in "both roots greater than k".
Using $\alpha\beta\gamma = \frac{d}{a}$ for a cubic; it is $-\frac{d}{a}$ .
Ignoring the sign of a in location-of-roots conditions.

Frequently Asked Questions

What is α² + β² in terms of the coefficients?

α² + β² = (α + β)² − 2αβ = (b² − 2ac)/a².

What is a symmetric function of roots?

An expression in the roots that does not change when the roots are interchanged, such as α + β or α²β + αβ².

How do you form an equation with reciprocal roots?

Replace x by 1/x. For ax² + bx + c = 0, the new equation is cx² + bx + a = 0.

When does a number k lie between the roots of a quadratic?

For ax² + bx + c with a > 0, when f(k) < 0. In general, when a·f(k) < 0.

What is the product of the roots of a cubic equation?

For ax³ + bx² + cx + d = 0, the product of the roots is −d/a.