Polynomials

Question 1

How do you use algebra to find the other roots of a polynomial given at least one root?

Answer 1

Write your given roots as brackets multilied by the unknown brackets, then use common sense

Question 2

If a polynomial has a complex route (a+bi) what do we know another route must be?

Answer 2

The complex conjugate (a-bi)

Question 3

What does a root of a polynomial look like in bracket form?

Answer 3

(x-(a+bi)) or (x-(a))

Question 4

How do you solve a polynomial equation with multiple unknown constants given some of its roots?

Answer 4

Sub in all of its roots to form simultaneous equations and solve them to find the constants

Note you can also write your roots in bracket form and multiply the brackets out, then use common sense to find the unknown brackets

Question 5

What is vieta’s formula in general?

Answer 5

• sum of roots in single pairs = -b/a
• sum of roots in double pairs = c/a
• sum of roots in triple pairs = -d/a
• sum of the roots in quad pairs = e/a

Where polynomial : aⁿ + b^n-1 + c^n-2 + d^n-3 + e^n-4 where the lowest power is n-1

Note quadratics only have up to double pairs and cubics only have up to triple pairs

Question 6

What are the roots of a polynomial called?

Answer 6

α β γ δ

Question 7

What would you do if asked to find atypical roots of polynomials? e.g 1/α + 1/β

Answer 7

• First find all vieta’s formulas for your polynomial
• Then rearrange and sub in your vietas formulas

Note sometimes you may have to use one of the vieta’s identities

Question 8

How do you deal with finding the roots of a polynomial when they are given interms of another polynomials roots? e.g a polynomial has roots α and β fnd a polynomial that has roots 2α and 2β

Answer 8

• First find all the vieta’s formulas you can for the original polynomial
• Then find in terms of αβγδ the sum of root pairs for the new polynomial (may need to use the identities)
• Write the sums of roots of the new polynomial in terms of vieta’s formulas for the original polynomial make sure the equal these too -b/a and c/a etc where a b c… are coeffiecients of the new polynomial
• Then sub in and solve for the coefficients of the new polynomial, a = 1 for the new polynomial

Question 9

What are the identities for quadratic roots?

Answer 9

α² + β² = (α+β)² - 2αβ

α³ + β³ - 3αβ(α+β)

Question 10

What are the identities for cubic roots?

Answer 10

α²+β²+γ² = (α+β+γ)² - 2(αβ+βγ+γα)

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

Question 11

what are the identities for quartic roots?

Answer 11

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