Roots,Roots Everywhere!

Algebra Level 3

Suppose roots of x 3 + 8 x 2 + 4 x 11 x^{3}+8x^{2}+4x-11 are α \alpha , β \beta and γ \gamma . And roots of x 3 + z x 2 + f x + g x^{3}+zx^{2}+fx+g are α + β \alpha + \beta , α + γ \alpha+ \gamma and β + γ \beta+ \gamma .Then, Find z × f × g z×f×g


The answer is 46784.

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1 solution

x 3 + 8 x 2 + 4 x 11 x^{3}+8x^{2}+4x-11
R o o t s α , β , γ Roots-\alpha,\beta,\gamma
B y V i e t a s F o r m u l a By Vieta's Formula
α + β + γ = 8 \alpha+\beta+\gamma=-8
α β γ = 11 \alpha\beta\gamma=11
α β + β γ + γ α = 4 \alpha\beta+\beta\gamma+\gamma\alpha=4
A n o t h e r P o l y n o m i a l Another Polynomial
x 3 + z x 2 + f x + g x^{3}+zx^{2}+fx+g
R o o t s ( α + β Roots-(\alpha + \beta ), ( α + γ \alpha+ \gamma ,) ( β + γ \beta+ \gamma )
A g a i n V i e t a s F o r m u l a Again Vieta's Formula
2 ( α + β + γ ) = z 2(\alpha+\beta+\gamma)=-z
z = 16 z=\boxed{16}
( α + β (\alpha+\beta ( α + γ (\alpha+\gamma ( β + γ (\beta+\gamma ) = g =-g
= = ( 8 γ (-8-\gamma ( 8 β (-8-\beta ( 8 α (-8-\alpha )= g -g
g = 43 g=\boxed{43}
( α + β (\alpha+\beta ) ( α + γ (\alpha+\gamma )+ ( α + γ (\alpha+\gamma ) ( β + γ (\beta+\gamma )+ ( α + β (\alpha+\beta ) ( β + γ (\beta+\gamma ) = f =f
= = α 2 + β 2 + γ 2 + 3 ( α β \alpha^{2}+\beta^{2}+\gamma^{2}+3(\alpha \beta + α γ \alpha \gamma + β γ \beta \gamma )
f = 68 f=\boxed{68}
N o w , Now,
z × g × f = 16 × 43 × 68 = 46784 z×g×f=16×43×68=\boxed{46784}


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