Core Pure 4: Finding Roots Of Polynomials

Question 1

If α and β are roots if the equation ax^2+bx+c = 0,

What is:
- α+β
- αβ

Answer 1

α+β = -b/a

αβ = c/a

Question 2

What is the layout for ‘ax^2 + bx + c’ factorised

Answer 2

ax^2 +bx + c ≡ a (x-α) (x-β)

Question 3

If α, β and γ are roots if the equation ax^3+bx^2+cx + d = 0,

What is:
- α+β+γ
- αβ + βγ + γα
- αβγ

Answer 3

α+β+γ = Σα = -b/a

αβ + βγ + γα = Σαβ = c/a

αβγ = -d/a

Question 4

What is the layout for ‘
Ax^3 + bx^2 + cx + d’ factorised

Answer 4

ax^3 + bx^2 + cx + d ≡ a (x-α) (x-β) (x-γ)

Question 5

If α, β, γ and δ are roots if the equation ax^4+bx^3+cx^2 + dx + e = 0,

What is:
- α+β+γ+δ
- αβ + αγ + αδ + βγ + βδ + γδ
- αβγ + αβδ + αγδ + βγδ
- αβγδ

Answer 5

-α+β+γ+δ = Σα = -b/a

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

αβγ + αβδ + αγδ + βγδ = Σαβγ = -d/a

αβγδ = e/a

Question 6

What is the layout for ‘ax^4 + bx^3 + cx^2 + dx + e’ factorised

Answer 6

ax^4 + bx^3 + cx^2 + dx + e ≡ a (x-α) (x-β) (x-γ) (x -δ)

Question 7

Rule for the Reciprocal of a:

• Quadratic
  1/α + 1/β
• Cubic
  1/α + 1/β + 1/γ
• Quartic
  1/α + 1/β + 1/γ + 1/δ

Answer 7

Quadratic:
1/α + 1/β = α+β/αβ

Cubic:
1/α + 1/β + 1/γ = αβ+βγ+γα / αβγ

Quartic:
1/α + 1/β + 1/γ + 1/δ = αβγ+βγδ+γδα+δαβ / αβγδ

Question 8

Rules of the sum of squares for a Quadratic

α^2 + β^2

Answer 8

α^2 + β^2 = (α+β)^2 - 2αβ

Question 9

Rules of the sum of squares for a Cubic

α^2 + β^2 + γ^2

Answer 9

α^2 + β^2 + γ^2 = (α+β+γ)^2 - 2(αβ+βγ+γα)

Question 10

Rules of the sum of squares for a Quartic

α^2 + β^2 + γ^2 + δ^2

Answer 10

α^2 + β^2 + γ^2 + δ^2 = (α+β+γ+δ)^2 -2(αβ + αγ + αδ + βγ + βδ + γδ)

Question 11

Rules of the sum of cubes for a Quadratic

α^3 + β^3

Answer 11

α^3 + β^3 = (α+β)^3 - 3αβ (α+β)

Question 12

Rules of the sum of cubes for a Cubuc

α^3 + β^3 + γ^3

Answer 12

α^3 + β^3 + γ^3 = (α+β+γ)^3 - 3(α+β+γ)(αβ+βγ+ γα) + 3 αβγ