Pure 1.4 - Roots of polynomials

Question 1

Describe the relationships between the roots of a quadratic equation and its coefficients.

Answer 1

If α and β are the roots of the equation 𝑎𝑥²+𝑏𝑥+𝑐=0, then:
α+β=-(𝑏/𝑎)
αβ=𝑐/𝑎

Question 2

Describe the relationships between the roots of a cubic equation and its coefficients.

Answer 2

If α, β and γ are the roots of the equation 𝑎𝑥³+𝑏𝑥²+𝑐𝑥+𝑑=0, then:
α+β+γ=-(𝑏/𝑎)
αβ+βγ+γα=𝑐/𝑎
αβγ=-(𝑑/𝑎)

Question 3

Describe the relationships between the roots of a quartic equation and its coefficients.

Answer 3

If α, β, γ and δ are the roots of the equation 𝑎𝑥⁴+𝑏𝑥³+𝑐𝑥²+𝑑𝑥+𝑒=0, then:
α+β+γ+δ=-(𝑏/𝑎)
αβ+βγ+γδ+δα+αγ+βδ=𝑐/𝑎
αβγ+αβδ+αγδ+βγδ=-(𝑑/𝑎)
αβγδ=𝑒/𝑎

Question 4

Recall the rules for reciprocals of quadratic, cubic and quartic polynomials, relating to their roots.

Answer 4

1/α+1/β=(α+β)/αβ
1/α+1/β+1/γ=(αβ+βγ+γα)/αβγ
1/α+1/β+1/γ+1/δ=(αβγ+αβδ+αγδ+βγδ)/αβγδ

Question 5

Recall the rules for the products of powers of quadratic, cubic and quartic polynomials, relating to their roots.

Answer 5

αⁿ×βⁿ=(αβ)ⁿ
αⁿ×βⁿ×γⁿ=(αβγ)ⁿ
αⁿ×βⁿ×γⁿ×δⁿ=(αβγδ)ⁿ

Question 6

Recall the rules for the sums of the squares of the roots of quadratic, cubic and quartic polynomials.

Answer 6

α²+β²=(α+β)²-2αβ
α²+β²+γ²=(α+β+γ)²-2(αβ+βγ+γα)
α²+β²+γ²+δ²=(α+β+γ+δ)²-2(αβ+βγ+γδ+δα+αγ+βδ)

Question 7

Recall the rules for the sums of the cubes of the roots of quadratic and cubic polynomials.

Answer 7

α³+β³=(α+β)³-3αβ(α+β)

α³+β³+γ³=(α+β+γ)³-3(αβ+βγ+γα)(α+β+γ)+3αβγ