Roots of Polynomials

Question 1

Symbols used for roots

Answer 1

α - alpha
β - beta
γ - gamma
δ - delta

Question 2

Rules for quadratic roots
ax^2 + bx + c = 0

Answer 2

Sum of roots = -(b/a)
Sum of all possible products of pairs of roots = (c/a)

Question 3

Rules for cubic roots
ax^3 + bx^2 + cx + d = 0

Answer 3

Sum of roots = -(b/a)
Sum of all possible products of pairs of roots = (c/a)
Sum of all possible products of triples of roots = -(d/a)

Question 4

Rules for quartic roots
ax^4 + bx^3 + cx^2 + dx + e = 0

Answer 4

Sum of roots = -(b/a)
Sum of all possible products of pairs of roots = (c/a)
Sum of all possible products of triples of roots = -(d/a)
Sum of all possible products of quadruples of roots = (e/a)

Question 5

1 1
– + –
α β

Answer 5

Multiply the two to have a common denominator and add
Will end up with:
.α + β
.——–
αβ

Question 6

α^2 + β^2
For a quadratic

Answer 6

(α+β)^2 - 2αβ
The expansion of (α+β)^2 is α^2 + 2αβ + β^2
Expand and simplify the same way for any others

Question 7

When you know α and β but not a,b and c in a quadratic

Answer 7

1.Do α + β for -(b/a) and αβ for (c/a)
2.Choose a value of a and find b and c for that a to find the ratio a:b:c (Use a = 1)
3. Write as ax^2 + bx + c = 0
4. Multiply a, b and c by the highest denominator to get integers and that equals zero

Question 8

1 1 1
– + – + –
α β γ

Answer 8

Use the substitution for the cubic for x = 1 / u. Let the new cubic have roots x, y, z. The sum of the roots x+ y + z = 1/α + 1/β + 1/γ

Question 9

1 1 1 1
– + – + – + –
α β γ 𝛿

Answer 9

Use the substitution for the quartic for x = 1 / u. Let the new cubic have roots w, x, y, z. The sum of the roots w + x + y + z = 1/α + 1/β + 1/γ + 1/𝛿

Question 10

How to go from polynomial with roots αβγ𝛿 to the new quadratic formed when the new roots are those changed in the same way

Answer 10

Let x = the symbols
w = the change to each root with x
Rearrange to find x
Substitute in place of x in the given polynomial
Leave in terms of w

Question 11

α^3 + β^3 + γ^3

Answer 11

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