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MATH / The polynomial f(x)=x³−3x²−4x+4 has three real roots r₁, r₂, and r₃. Let g(x)=x³+ax²+bx+c be the polynomial which has roots s₁, s₂, and s₃, where …z=(−1+i√(3))/(2). Find the real p…

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problem

The polynomial f(x)=x3−3x2−4x+4f(x)=x^3-3x^2-4x+4 has three real roots r1r_1, r2r_2, and r3r_3. Let g(x)=x3+ax2+bx+cg(x)=x^3+ax^2+bx+c be the polynomial which has roots s1s_1, s2s_2, and s3s_3, where \begin{align*} s_1 &= r_1+r_2z+r_3z^2, \\ s_2 &= r_1z+r_2z^2+r_3, \\ s_3 &= r_1z^2+r_2+r_3z, \end{align*}and z=−1+i32z=\frac{-1+i\sqrt3}2. Find the real part of the sum of the coefficients of g(x)g(x).
Plain-text mathematical notation (without MathML)
The polynomial f(x)=x³−3x²−4x+4 has three real roots r₁, r₂, and r₃.  Let g(x)=x³+ax²+bx+c be the polynomial which has roots s₁, s₂, and s₃, where
\begin{align*}
s_1 &= r_1+r_2z+r_3z^2, \\
s_2 &= r_1z+r_2z^2+r_3, \\
s_3 &= r_1z^2+r_2+r_3z,
\end{align*}and z=(−1+i√(3))/(2).  Find the real part of the sum of the coefficients of g(x).
Original LaTeX notation
The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$.  Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where
\begin{align*}
s_1 &= r_1+r_2z+r_3z^2, \\
s_2 &= r_1z+r_2z^2+r_3, \\
s_3 &= r_1z^2+r_2+r_3z,
\end{align*}and $z=\frac{-1+i\sqrt3}2$.  Find the real part of the sum of the coefficients of $g(x)$.

level

Level 5

type

Intermediate Algebra

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task_id
4085dd6a-4002-5d8f-9dd9-3c82b3833a78
task_key
intermediate~5falgebra--test--4085dd6a-4002-5d8f-9dd9-3c82b3833a78
task_revision_id
1
config / split
intermediate_algebra / test

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