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AS-Level Further Maths

Revision Session 1

  • Roots of Polynomials

Revision Session 2

  • Proof by induction

Revision Session 3

  • Momentum and Impulse

Revision Session 4

  • Review and Hooke’s Law

​

2 of 15

From the specification and teaching guidance

3 of 15

Roots of a polynomial

General Rules

axn + bxn-1 + cxn-2 + dxn-3 + … + constant = 0

Quartic (ie. n=4):

b

a

α + β + γ + δ = -

c

a

Σαβ =

Σαβγ = -

αβγδ =

d

a

constant

a

e

a

=

Cubic (ie. n=3):

b

a

α + β + γ = -

c

a

Σαβ =

αβγ = -

constant

a

d

a

= -

Quadratic (ie. n=2):

b

a

α + β = -

αβ =

constant

a

c

a

=

Learn the pattern!

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Roots of a polynomial

Cubic (ie. n=3):

b

a

α + β + γ = -

c

a

Σαβ =

αβγ = -

constant

a

d

a

= -

5 of 15

From the specification and teaching guidance

6 of 15

Roots of a polynomial – Related Expressions

Cubic (ie. n=3):

b

a

α + β + γ = -

c

a

Σαβ =

αβγ = -

constant

a

d

a

= -

(αβ + αγ + βγ)2 = (αβ)2 + (αγ)2 + (βγ)2 + α2βγ + α2βγ + αβ2γ + αβ2γ + αβγ2 + αβγ2

​

(αβ + αγ + βγ)2 = (αβ)2 + (αγ)2 + (βγ)2 + 2αβγ(α + β + γ)

​

(αβ)2 + (αγ)2 + (βγ)2 = (αβ + αγ + βγ)2 - 2αβγ(α + β + γ)

​

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Roots of a polynomial – Related Expressions

Cubic (ie. n=3):

b

a

α + β + γ = -

c

a

Σαβ =

αβγ = -

constant

a

d

a

= -

(αβ + αγ + βγ)2 = (αβ)2 + (αγ)2 + (βγ)2 + α2βγ + α2βγ + αβ2γ + αβ2γ + αβγ2 + αβγ2

​

(αβ + αγ + βγ)2 = (αβ)2 + (αγ)2 + (βγ)2 + 2αβγ(α + β + γ)

​

(αβ)2 + (αγ)2 + (βγ)2 = (αβ + αγ + βγ)2 - 2αβγ(α + β + γ)

​

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Roots of a polynomial

9 of 15

Roots of a polynomial - Transformations

x3 – 2x2 + 3x – 4 = 0 has roots α, β and γ.

​

Find equations of polynomials with roots:

a) 2α, 2β and 2γ

Cubic (ie. n=3):

b

a

α + β + γ = -

c

a

Σαβ =

αβγ = -

constant

a

d

a

= -

Method 1

Method 2

10 of 15

Roots of a polynomial - Transformations

x3 – 2x2 + 3x – 4 = 0 has roots α, β and γ.

​

Find equations of polynomials with roots:

a) 2α, 2β and 2γ

Cubic (ie. n=3):

b

a

α + β + γ = -

c

a

Σαβ =

αβγ = -

constant

a

d

a

= -

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Roots of a polynomial – Number of Roots

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Roots of a polynomial – Number of Roots

13 of 15

Roots of a polynomial – Number of Roots

14 of 15

From the specification and teaching guidance

15 of 15

AS-Level Further Maths

Revision Session 1

  • Roots of Polynomials

Revision Session 2

  • Proof by induction

Revision Session 3

  • Momentum and Impulse

Revision Session 4

  • Review and Hooke’s Law

​