3.1.1 Roots of Polynomials

Understanding the Roots and Coefficients of Polynomial Equations

The roots of a polynomial are the values that satisfy the equation $f(x) = 0$ .

For any polynomial equation, there is a direct relationship between its roots and its coefficients. These relationships are derived from the expanded form of the polynomial and are essential for evaluating expressions involving roots.

General Form of a Polynomial

A polynomial of degree $n$ can be written as:

f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0

where a $_n \neq 0$ .

The roots of the polynomial are denoted

\alpha, \beta, \gamma, \dots

and the coefficients $a_0, a_1, \dots$ determine the polynomial's structure.

Relationships Between Roots and Coefficients

For a polynomial equation of degree $nn$ :

a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 = 0,

if the roots are $\alpha, \beta, \gamma, \dots$ , then:

The sum of roots (denoted): $S_1$

S_1 = \alpha + \beta + \gamma + \dots = -\frac{a_{n-1}}{a_n}

The sum of products of roots taken two at a time (denoted): $S_2$

S_2 = \alpha\beta + \alpha\gamma + \beta\gamma + \dots = \frac{a_{n-2}}{a_n}

The product of all roots (denoted): $P$

P = \alpha\beta\gamma\dots = (-1)^n \frac{a_0}{a_n}

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Example

For a cubic equation

ax^3 + bx^2 + cx + d = 0

\alpha + \beta + \gamma = -\frac{b}{a}

\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}

\alpha\beta\gamma = -\frac{d}{a}

Example Problems

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Evaluate

\alpha^2 + \beta^2 + \gamma^2

From the identity:

\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)

we substitute:

S_1 = \alpha + \beta + \gamma, \quad S_2 = \alpha\beta + \beta\gamma + \gamma\alpha.

For a cubic polynomial

ax^3 + bx^2 + cx + d = 0

\alpha^2 + \beta^2 + \gamma^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right)

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Evaluate

\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}

Using the identity:

\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\alpha\beta + \beta\gamma + \gamma\alpha}{\alpha\beta\gamma}

we substitute:

S_2 = \alpha\beta + \beta\gamma + \gamma\alpha, \quad P = \alpha\beta\gamma

For a cubic polynomial $ax^3 + bx^2 + cx + d = 0$

\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\frac{c}{a}}{-\frac{d}{a}} = -\frac{c}{d}

infoNote

Evaluate

(3 + \alpha)(3 + \beta)(3 + \gamma)

This expands to:

(3 + \alpha)(3 + \beta)(3 + \gamma) = 27 + 9(\alpha + \beta + \gamma) + 3(\alpha\beta + \beta\gamma + \gamma\alpha) + \alpha\beta\gamma

Substitute:

S_1 = \alpha + \beta + \gamma

S_2 = \alpha\beta + \beta\gamma + \gamma\alpha

P = \alpha\beta\gamma

For a cubic polynomial $ax^3 + bx^2 + cx + d = 0$

(3 + \alpha)(3 + \beta)(3 + \gamma) = 27 + 9\left(-\frac{b}{a}\right) + 3\left(\frac{c}{a}\right) + \left(-\frac{d}{a}\right).

infoNote

Evaluate

\alpha^3 + \beta^3 + \gamma^3

Using the identity:

\alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)\left(\alpha^2 + \beta^2 + \gamma^2 - \alpha\beta - \beta\gamma - \gamma\alpha\right) + 3\alpha\beta\gamma

we substitute:

S_1, S_2, \text{ and } P

For a cubic polynomial $ax^3 + bx^2 + cx + d = 0$

\alpha^3 + \beta^3 + \gamma^3 = \left(-\frac{b}{a}\right)\left[\left(-\frac{b}{a}\right)^2 - 3\left(\frac{c}{a}\right)\right] + 3\left(-\frac{d}{a}\right)

Note Summary

infoNote

Common Mistakes

Sign errors: Forgetting the negative signs in the relationships, e.g., . $S_1 = -\frac{b}{a}$
Misinterpreting product relationships: Confusing (sum of products of roots taken two at a time) with (product of all roots).
Expanding incorrectly: Errors in algebra when substituting into expanded expressions like . $(3 + \alpha)(3 + \beta)(3 + \gamma)$
Mixing formulas: Using cubic formulas for non-cubic polynomials or vice versa.
Arithmetic slips: Basic calculation errors when substituting coefficients.

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Key Formulas

The sum of roots: $S_1 = -\frac{a_{n-1}}{a_n}$
The sum of products of roots (two at a time):. $S_2 = \frac{a_{n-2}}{a_n}$
Product of roots: $P = (-1)^n \frac{a_0}{a_n}$
The sum of squares of roots: $\alpha^2 + \beta^2 + \gamma^2 = S_1^2 - 2S_2$
Expanded expressions: $(c + \alpha)(c + \beta)(c + \gamma) = c^3 + S_1c^2 + S_2c + P$

Roots of Polynomials Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

3.1.1 Roots of Polynomials

Understanding the Roots and Coefficients of Polynomial Equations

General Form of a Polynomial

Relationships Between Roots and Coefficients

The sum of roots (denoted): $S_1$

The sum of products of roots taken two at a time (denoted): $S_2$

The product of all roots (denoted): $P$

Example

Example Problems

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Roots of Polynomials For their A-Level Exams.

Other Revision Notes related to Roots of Polynomials you should explore

Roots of Polynomials Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

3.1.1 Roots of Polynomials

Understanding the Roots and Coefficients of Polynomial Equations

General Form of a Polynomial

Relationships Between Roots and Coefficients

The sum of roots (denoted): S1S_1S1​

The sum of products of roots taken two at a time (denoted): S2S_2S2​

The product of all roots (denoted): PPP

Example

Example Problems

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Roots of Polynomials For their A-Level Exams.

Other Revision Notes related to Roots of Polynomials you should explore

The sum of roots (denoted): $S_1$

The sum of products of roots taken two at a time (denoted): $S_2$

The product of all roots (denoted): $P$