Roots of Polynomials (Edexcel A-Level Further Mathematics): Revision Notes

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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 equationf(x)=0 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 nn can be written as:

f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0

where an≠0_n \neq 0.

The roots of the polynomial are denoted

α,β,γ,…\alpha, \beta, \gamma, \dots

and the coefficients a0,a1,…a_0, a_1, \dots determine the polynomial's structure.

Relationships Between Roots and Coefficients

For a polynomial equation of degree nnnn:

anxn+an−1xn−1+⋯+a0=0,a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 = 0,

if the roots are α,β,γ,…\alpha, \beta, \gamma, \dots, then:

The sum of roots (denoted): S1S_1

S1=α+β+γ+⋯=−an−1anS_1 = \alpha + \beta + \gamma + \dots = -\frac{a_{n-1}}{a_n}

The sum of products of roots taken two at a time (denoted): S2S_2

S2=αβ+αγ+βγ+⋯=an−2anS_2 = \alpha\beta + \alpha\gamma + \beta\gamma + \dots = \frac{a_{n-2}}{a_n}

The product of all roots (denoted): PP

P=αβγ⋯=(−1)na0anP = \alpha\beta\gamma\dots = (-1)^n \frac{a_0}{a_n}
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Example

For a cubic equation

ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0
α+β+γ=−ba\alpha + \beta + \gamma = -\frac{b}{a}αβ+βγ+γα=ca\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}αβγ=−da\alpha\beta\gamma = -\frac{d}{a}

Example Problems

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Evaluate

α2+β2+γ2\alpha^2 + \beta^2 + \gamma^2

From the identity:

α2+β2+γ2=(α+β+γ)2−2(αβ+βγ+γα)\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)

we substitute:

S1=α+β+γ,S2=αβ+βγ+γα.S_1 = \alpha + \beta + \gamma, \quad S_2 = \alpha\beta + \beta\gamma + \gamma\alpha.

For a cubic polynomial

ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0α2+β2+γ2=(−ba)2−2(ca)\alpha^2 + \beta^2 + \gamma^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right)
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Evaluate

1α+1β+1γ\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}

Using the identity:

1α+1β+1γ=αβ+βγ+γααβγ\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\alpha\beta + \beta\gamma + \gamma\alpha}{\alpha\beta\gamma}

we substitute:

S2=αβ+βγ+γα,P=αβγS_2 = \alpha\beta + \beta\gamma + \gamma\alpha, \quad P = \alpha\beta\gamma

For a cubic polynomial ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0

1α+1β+1γ=ca−da=−cd\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\frac{c}{a}}{-\frac{d}{a}} = -\frac{c}{d}
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Evaluate

(3+α)(3+β)(3+γ)(3 + \alpha)(3 + \beta)(3 + \gamma)

This expands to:

(3+α)(3+β)(3+γ)=27+9(α+β+γ)+3(αβ+βγ+γα)+αβγ(3 + \alpha)(3 + \beta)(3 + \gamma) = 27 + 9(\alpha + \beta + \gamma) + 3(\alpha\beta + \beta\gamma + \gamma\alpha) + \alpha\beta\gamma

Substitute:

S1=α+β+γS_1 = \alpha + \beta + \gammaS2=αβ+βγ+γαS_2 = \alpha\beta + \beta\gamma + \gamma\alphaP=αβγP = \alpha\beta\gamma

For a cubic polynomial ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0

(3+α)(3+β)(3+γ)=27+9(−ba)+3(ca)+(−da).(3 + \alpha)(3 + \beta)(3 + \gamma) = 27 + 9\left(-\frac{b}{a}\right) + 3\left(\frac{c}{a}\right) + \left(-\frac{d}{a}\right).
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Evaluate

α3+β3+γ3\alpha^3 + \beta^3 + \gamma^3

Using the identity:

α3+β3+γ3=(α+β+γ)(α2+β2+γ2−αβ−βγ−γα)+3αβγ\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:

S1,S2, and PS_1, S_2, \text{ and } P

For a cubic polynomial ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0

α3+β3+γ3=(−ba)[(−ba)2−3(ca)]+3(−da)\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

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Common Mistakes

  1. Sign errors: Forgetting the negative signs in the relationships, e.g., . S1=−baS_1 = -\frac{b}{a}

  2. Misinterpreting product relationships: Confusing (sum of products of roots taken two at a time) with (product of all roots).

  3. Expanding incorrectly: Errors in algebra when substituting into expanded expressions like . (3+α)(3+β)(3+γ)(3 + \alpha)(3 + \beta)(3 + \gamma)

  4. Mixing formulas: Using cubic formulas for non-cubic polynomials or vice versa.

  5. Arithmetic slips: Basic calculation errors when substituting coefficients.

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Key Formulas

  1. The sum of roots: S1=−an−1anS_1 = -\frac{a_{n-1}}{a_n}

  2. The sum of products of roots (two at a time):. S2=an−2anS_2 = \frac{a_{n-2}}{a_n}

  3. Product of roots: P=(−1)na0anP = (-1)^n \frac{a_0}{a_n}

  4. The sum of squares of roots: α2+β2+γ2=S12−2S2\alpha^2 + \beta^2 + \gamma^2 = S_1^2 - 2S_2

  5. Expanded expressions: (c+α)(c+β)(c+γ)=c3+S1c2+S2c+P(c + \alpha)(c + \beta)(c + \gamma) = c^3 + S_1c^2 + S_2c + P

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