5 - roots of polynomials

Question 1

if ⍺ β are the roots of a quadratic

Answer 1

x² -(⍺ + β)x + ⍺β

Question 2

if ⍺ is a root and is complex

Answer 2

the other root is ⍺*

Question 3

if ⍺ β 𝛾 are the roots of a cubic

Answer 3

x³ -(⍺ + β+ 𝛾)x ² +( ⍺β +⍺𝛾+ β𝛾)x - ⍺β𝛾

Question 4

roots of a cubic

Answer 4

• 3 separate real
• 1 real + 2 complex conjugate pairs
• 1 real + 1 repeated

Question 5

ax² + bx + c

Answer 5

-b/a = ⍺+ β
c/a = ⍺β

Question 6

ax³ + bx² + cx + d

Answer 6

-b/a = ⍺+ β + 𝛾
c/a = ⍺β + β𝛾 + ⍺𝛾
-d/a = ⍺β𝛾

Question 7

ax⁴ + bx³ + cx² + dx + e

Answer 7

-b/a = ⍺+ β + 𝛾 + 𝛿
c/a = ⍺β + ⍺𝛾 + ⍺𝛿+ β𝛾 + β𝛿 + 𝛾𝛿
-d/a = ⍺β𝛾 + ⍺β𝛿 + ⍺𝛾𝛿 + β𝛾𝛿
e/a = ⍺β𝛾𝛿

Question 8

using substitution to find new eq

Answer 8

3x^3 -x^2 + 2x + 5 = 0 has roots p q r
find a cubic with roots (p-2) (q-2) (r-2)
u = x-2
x = u+ 2 < sub into original eq