help (1 Viewer)

[currysauce]

this should be relatively easy for you guys... if you can work these 2 qu's out ur champions... i have tried and its not working for me.

Polymonials and Roots, 2 questions.

1. Two roots of x^3 + mx^2 + 15x - 7 = 0 are equal and rational. Find m.

2. Two roots of x^3 + ax^2 + bx - 5 = 0 are equal to 4 and - 2. Find the values of a and b.

 

[wogboy]

1. Suppose the roots are a, a, and b.

b*a^2 = 7 (product of roots)
a^2 + 2*ab = 15 (sum of coupled root products)

-> a^2 + 14/a = 15
-> a^3 - 15a + 14 = 0

by the rational root theorem, if there exists a rational root for the above polynomial, it must be a factor of 14 -> a = 1 is the only rational root.

-> a = 1
-> b = 7

-> m = -(2a + b) = -9 (sum of roots)

2. Let the third root be r.

-8r = 5 (product of roots)
r = -5/8

-> a = -(4 - 2 - 5/8) = -11/8
-> b = 4*(-2) + 4*(-5/8) + (-2)*(-5/8) = -37/4

 

[currysauce]

1 more qu

one more questions please!

Solve

6x^4 + 5x^3 - 24x^2 - 15x + 18 = 0 if the sum of two of its roots is zero

 

[Estel]

6x^4 + 5x^3 - 24x^2 - 15x + 18 = 0

let the roots be a, -a, b and c
b + c = -5/6
-a^2b - a^2c = -a^2(b+c) = 5/2
i.e. a^2 = 3
then two roots are rt3 and -rt3
-a^2bc = 3
bc = -1
b(b + 5/6) = 1
b = 2/3
c = -3/2

so roots are 2/3, -3/2, rt3 and -rt3