SOLUTION: The equation of y=x³+4x²+x-6 has roots α, β and γ. Find the value of (i) α²+β²+γ² (ii) α²β²γ² (iii) (α+β)(β+γ)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The equation of y=x³+4x²+x-6 has roots α, β and γ. Find the value of (i) α²+β²+γ² (ii) α²β²γ² (iii) (α+β)(β+γ)      Log On


   

Question 1132057: The equation of y=x³+4x²+x-6 has roots α, β and γ. Find the value of
(i) α²+β²+γ²
(ii) α²β²γ²
(iii) (α+β)(β+γ)(γ+α)

Found 2 solutions by Boreal, MathLover1:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
x^3+4x^2+x-6=0
with synthetic division, using -3
1-----4----1---- -6
1-----1--- -2-----0
(x+3)(x^2+x-2)=0, so -3 is a root
(x+3)(x+2)(x-1)=0
roots are -3, -2, and +1
(i) is 14
(ii) is 36
(iii) is (-5)(-1)(-2)=-10
graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E3%2B4x%5E2%2Bx-6%29

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

The equation of y=x%5E3%2B4x%5E2%2Bx-6 has roots alpha, beta and gamma}.
Find the value of
(i) alpha%5E2%2Bbeta%5E2%2Bgamma%5E2
(ii) alpha%5E2%2Abeta%5E2%2Agamma%5E2
(iii)
%28alpha-beta%29%28beta%2Bgamma%29%28gamma%2Balpha%29

y=x%5E3%2B4x%5E2%2Bx-6.......factor completely

y=x%5E3%2B3x%5E2%2B2x%5E2%2B6x-x%5E2-3x-2x-6

y=%28x%5E3%2B3x%5E2%29%2B%282x%5E2%2B6x%29-%28x%5E2%2B3x%29-%282x%2B6%29

y=x%5E2%28x%2B3%29%2B2x%28x%2B3%29-x%28x%2B3%29-2%28x%2B3%29

y+=+%28x%5E2%2B2x-x-2%29%28x+%2B+3%29

y+=+%28x+-+1%29+%28x+%2B+2%29+%28x+%2B+3%29

solutions:
x=1
x=-2
x=-3
=> let's alpha=1, beta=-2 and gamma=-3}
then,
(i)
alpha%5E2%2Bbeta%5E2%2Bgamma%5E2
1%5E2%2B%28-2%29%5E2%2B%28-3%29%5E2
1%2B4%2B9
14



(ii)
alpha%5E2%2Abeta%5E2%2Agamma%5E2
1%5E2%2A%28-2%29%5E2%2A%28-3%29%5E2
1%2A4%2A9
36

(iii)

%28alpha-beta%29%28beta%2Bgamma%29%28gamma%2Balpha%29
%281-%28-2%29%29%28-2-3%29%28-3%2B1%29
%281%2B2%29%28-5%29%28-2%29
%283%29%2810%29
30