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The cubic equation 2x³ + 5x² - 6 = 0 has roots a, ß,y.

The cubic equation 2x³ + 5x² - 6 = 0 has roots a, ß,y.-example-1
User Yrlec
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1 Answer

4 votes

Answer:

(a) 216x³ -216x² -53x -8

(b) 161/108

(c) 133/72

Explanation:

Given that α, β, and γ are the roots of the cubic 2x³ +5x² -6 = 0, you want ...

(a) a cubic with roots 1/α³, 1/β³, 1/γ³
(b) the value of 1/α⁶ +1/β⁶ + 1/γ⁶
(c) the value of 1/α⁹ +1/β⁹ + 1/γ⁹

General case

Given the monic cubic x³ +ax² +bx +c = 0, we know that a, b, c are related to the roots by ...

  • a = -α -β -γ
  • b = αβ +αγ +βγ
  • c = -αβγ

Then if 1/α³, 1/β³, 1/γ³ are the roots of x³ +a'x² +b'x +c' = 0, we can show algebraically that ...

a' = (b³ - 3 a b c + 3 c²)/c³

b' = (a³ - 3 a b + 3 c)/c³

c' = 1/c³

We can also show that ...

α² +β² +γ² = a² -2b

α³ +β³ +γ³ = 3(ab -c) -a³

(a) Inverse cubes

Using the above relations we find the coefficients of the required cubic can be found from a = 5/2, b = 0, c = -3:

a' = (b³ - 3 a b c + 3 c²)/c³ = -1

b' = (a³ - 3 a b + 3 c)/c³ = -53/216

c' = 1/c³ = -1/27

For the next parts, we will use this version of the desired cubic:

x³ -x² -(53/216)x -1/27 = 0

We can write this with integer coefficients as ...

216x³ -216x² -53x -8 = 0 . . . . . . equation of the required cubic

(b) Sum of -6th powers

The cubic we just found has roots 1/α³, 1/β³, 1/γ³, and we want the value of the sum of the squares of these roots: 1/α⁶ +1/β⁶ + 1/γ⁶.

For the purpose we can use the relation α² +β² +γ² = a² -2b, where the coefficients are the a' and b' we found above

1/α⁶ +1/β⁶ + 1/γ⁶ = (-1)² -2(-53/216) = 1 +106/216

1/α⁶ +1/β⁶ + 1/γ⁶ = 161/108

(c) Sum of -9th powers

As above, we need to find the sum of the cubes of the roots of the new cubic. This is given by the relation α³ +β³ +γ³ = 3(ab -c) -a³.

1/α⁹ +1/β⁹ + 1/γ⁹ = 3(-1(-53/216 -(-1/27)) -(-1)³

1/α⁹ +1/β⁹ + 1/γ⁹ = 133/72

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Additional comment

The first attachment shows the calculations using the formulas we found. The second attachment shows a machine solution to this problem. Note that the original equation has one real root and two complex roots.

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The cubic equation 2x³ + 5x² - 6 = 0 has roots a, ß,y.-example-1
The cubic equation 2x³ + 5x² - 6 = 0 has roots a, ß,y.-example-2
User Simone Avogadro
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