Y1, C4 - Roots of Polynomials

Question 1

What is the sum of the roots of a quadratic, cubic and quartic equation

Answer 1

α + β = -b / a
α + β + γ = -b / a
α + β + γ + δ = -b / a

Question 2

What is the sum of products of pairs of roots of a quadratic, cubic and quartic equation

Answer 2

αβ = c/a
αβ + αγ + βγ = c/a
αβ + αγ + αδ + βγ + βδ + γδ = c/a

Question 3

What is the sum of the products of triples of roots for a quadratic, cubic and quartic equation

Answer 3

N/A
αβγ = -d/a
αβγ + αγδ + αβδ + βγδ = -d/a

Question 4

What would the product of the all roots of a quartic

Answer 4

e / a

Question 5

What would be the product of all roots of a cubic

Answer 5

-d / a

Question 6

Find the sum and product of the roots of the quadratic x^2 + 2x + 3

Answer 6

Sum = -2
Product = 3

Question 7

How would you find the value of (1 / α) + (1 / β)

Answer 7

Add them = (α + β) / αβ
Sub in previously found values

Question 8

How would you find the value of α^2 + β^2

Answer 8

(α+β)^2 = α^2 + 2αβ + β^2
(α+β)^2 - 2αβ = α^2 + β^2
Sub in previously found values

Question 9

How can the sum of product pairs be written in sigma notation

Answer 9

Sigma(αβ)

Question 10

How can the sum of the roots be written in sigma notation

Answer 10

Sigma(α)

Question 11

How can (1 / α) + (1 / β) + (1 / γ) be written

Answer 11

(αβ + αγ + βγ) / αβγ

Question 12

What is the sum of products of quadruples

Answer 12

αβγδ = e/a

Question 13

What is the sigma notation for the sum of products of quadruples

Answer 13

Sigma(αβγδ)

Question 14

What comes after delta (δ) in the greek alphabet

Answer 14

Epsilon (ε)

Question 15

How can we write the sum of squares for quadratics, cubics and quartics in sigma notation?

Answer 15

α^2 + β^2 = Sigma(α)^2 - 2Sigma(αβ)
α^2 + β^2 + γ^2 = Sigma(α)^2 - 2Sigma(αβ)
α^2 + β^2 + γ^2 + δ^2= Sigma(α)^2 - 2Sigma(αβ)

Question 16

How can we write the sum of cubes for quadratic and cubics in sigma notation

Answer 16

α^3 + β^3 = Sigma(α)^3 - 3Sigma(αβ) * Sigma(α)
α^3 + β^3 + γ^3 = Sigma(α)^3 - 3Sigma(αβ) * Sigma(α) + 3αβγ

Question 17

What is the sum of reciprocals for the quartic roots

Answer 17

(Sigma(αβγ)) / αβγδ

Question 18

What would α^n * β^n * γ^n * δ^n equal

Answer 18

(αβγδ)^n
(Follows index laws)

Question 19

How would you find the value of (α + 3)(β + 3)(γ + 3) using sigma notation

Answer 19

Multiply terms out
= αβγ + 3Sigma(αβ) + 9Sigma(α) + 27
Plug in known values

Question 20

The quartic equation x^4 - 3x^3 + 15x + 1 = 0, has roots α, β, γ, δ. Find the equation with roots (2α+1),(2β+1), (2γ+1), (2δ+1)

Answer 20

Let w = 2x+1, rearrange to (w-1) / 2 = x
Substitute into equation as x
= ((w-1)^4 / 16) - 3((w-1)^3 / 2) + 15((w-1) / 2) + 1 = 0
Multiply through by 16
= (w-1)^4 = 6(w-1)^3 + 120(w-1) + 16 = 0
= w^4 - 10w^3 + 24w^2 + 98w -97 = 0

Question 21

What should you do if you don’t know the value of ‘a’

Answer 21

Set it to 1