Construct seven roots and combine conjugate pairs

Use the root and factor foundations

First complete polar powers, the general nth-root rule and conjugate factorisation. Apply these methods to seven roots. A diagram requires every root, a radius, labelled axes and the angular convention

Construct all seven directions

Find the radius and every distinct direction from the modulus and argument equations

1. 1Polar target Use the positive seventh-root radius \(\displaystyle \begin{gathered}z^7=-2187=3^7\operatorname{cis}\pi\\r=3\end{gathered}\)
2. 2Every root Retain the whole-turn term before dividing the argument \(\displaystyle z_j=3\operatorname{cis}\frac{(1+2j)\pi}{7},\quad j=0,\ldots,6\)
3. 3Arguments and spacing List the seven angles in increasing order \(\displaystyle \begin{gathered}\theta\in\left\{\frac\pi7,\frac{3\pi}7,\frac{5\pi}7,\pi,\right.\\\left.\frac{9\pi}7,\frac{11\pi}7,\frac{13\pi}7\right\}\\\Delta\theta=\frac{2\pi}{7}\end{gathered}\)

Give the common radius of these roots

Give the coefficient of π in the cyclic angular spacing

Which statement agrees with the construction?

Pair roots before expanding

Let ω=cis(2π/7) and take roots Rωʲ of z⁷=R⁷, R>0. The real root is R; the other six form three conjugate pairs, j and 7−j. Each pair contributes x²−2R cos(2jπ/7)x+R²

Work a scaled factor

For R=4, the pair with j=2 contributes x²−8cos(4π/7)x+16. Keep the trigonometric coefficient exact. The full product is (x−4) times all three conjugate quadratics; one quadratic alone is not the complete factorisation

Conjugate roots

\(\displaystyle r_{\pm}=4\operatorname{cis}\left(\pm\frac{4\pi}{7}\right)\)

Term

\(\displaystyle \cos\left(\frac{4\pi}{7}\right)x\)

Give the coefficient multiplying the displayed cosine term in the monic quadratic

Conjugate roots

\(\displaystyle r_{\pm}=4\operatorname{cis}\left(\pm\frac{4\pi}{7}\right)\)

Give the constant coefficient of the monic quadratic with these roots

Sum powers with the exceptional case visible

Pair conjugate powers in the full geometric sum, checking the exceptional exponent before dividing

1. 1Definitions Use a primitive seventh root of unity and a positive integer exponent \(\displaystyle \begin{gathered}\omega=\operatorname{cis}\frac{2\pi}{7},\quad m\in\mathbb Z_{>0}\\C_m=\sum_{j=1}^{3}\cos\frac{2jm\pi}{7}\end{gathered}\)
2. 2Pairing The conjugate terms give twice their real parts \(\displaystyle \sum_{j=0}^{6}\omega^{jm}=1+2C_m\)
3. 3Nonexceptional exponent The geometric ratio is not one \(\displaystyle \begin{gathered}7\nmid m\quad\Longrightarrow\quad\omega^m\ne1\\1+2C_m=0\\C_m=-\frac12\end{gathered}\)
4. 4Exceptional exponent Every term of the geometric sum is one, so do not use a zero denominator \(\displaystyle \begin{gathered}7\mid m\quad\Longrightarrow\quad\omega^m=1\\1+2C_m=7\\C_m=3\end{gathered}\)

For R=2 and m=3, give the sum of the powers of the six nonreal roots of z⁷=128.

Root equation

\(\displaystyle z^7=128=2^7\)

Power

\(\displaystyle m=7\)

Give the sum of the seventh powers over all seven roots

Construct and justify an Argand diagram, ungraded

Independently sketch all seven roots of z⁷=−2187. Label axes, radius, every argument in [0,2π), and the real root. Explain why the final gap also equals 2π/7, why every marked point solves the equation and why the circle is not itself a solution locus

Model: a complete finite construction

Draw a circle of radius 3 as a guide. Mark the arguments (1+2j)π/7 for j=0,…,6, with −3 on the negative real axis. The wrap-around gap is 2π+π/7−13π/7=2π/7. Raising each value to the seventh power gives 3⁷cis((1+2j)π)=−2187. The seven arguments are distinct modulo 2π, and a degree-seven polynomial has no further distinct roots. Only the marked vertices are solutions

Prove a factor and cosine identity, ungraded

Derive the factorisation by pairing conjugate roots, then prove both cases of the cosine-sum rule for every positive integer exponent. Explain what a numerical plot cannot establish

1. 1Root set Use the positive scale and a primitive seventh root of unity \(\displaystyle \begin{gathered}R>0,\quad\omega=\operatorname{cis}\frac{2\pi}{7}\\z_j=R\omega^j\quad(j=0,\ldots,6)\end{gathered}\)
2. 2Quadratics Derive the quadratic from each conjugate pair \(\displaystyle \begin{aligned}Q_j(x)&=x^2-2R\cos\frac{2j\pi}{7}\,x\\&\quad+R^2\qquad(j=1,2,3)\end{aligned}\)
3. 3Factor identity Prove the complete monic product \(\displaystyle x^7-R^7=(x-R)\prod_{j=1}^{3}Q_j(x)\)
4. 4Cosine sum Prove the displayed rule \(\displaystyle \begin{gathered}C_m=\sum_{j=1}^{3}\cos\frac{2jm\pi}{7},\quad m\in\mathbb Z_{>0}\\C_m=\begin{cases}3&7\mid m\\-\frac12&7\nmid m\end{cases}\end{gathered}\)

Model: use conjugation, completeness and a geometric sum

Use conjugate pairs for the real factorisation and a geometric sum for the cosine identity. A finite numerical check cannot prove either identity for every scale or exponent

1. 1Define the roots Use the positive scale and primitive root \(\displaystyle \begin{gathered}R>0,\quad\omega=\operatorname{cis}\frac{2\pi}{7}\\z_j=R\omega^j\quad(j=0,\ldots,6)\\\overline{z_j}=R\omega^{7-j}\end{gathered}\)
2. 2Multiply a conjugate pair Their sum is twice the real part and their product is the squared modulus \(\displaystyle \begin{aligned}(x-z_j)(x-\overline{z_j})&=Q_j(x)\\Q_j(x)&=x^2-2R\cos\frac{2j\pi}{7}\,x\\&\quad+R^2\end{aligned}\)
3. 3Complete the product The seven roots are distinct and both sides are monic of degree seven \(\displaystyle x^7-R^7=(x-R)\prod_{j=1}^{3}Q_j(x)\)
4. 4Define the cosine sum Conjugate terms have opposite imaginary parts \(\displaystyle \begin{gathered}C_m=\sum_{j=1}^{3}\cos\frac{2jm\pi}{7}\\1+2C_m=\sum_{j=0}^{6}\omega^{jm}\end{gathered}\)
5. 5When the exponent is not divisible by seven The nonzero denominator permits the geometric quotient \(\displaystyle \begin{gathered}q=\omega^m\ne1,\quad q^7=1\\\sum_{j=0}^{6}q^j=\frac{1-q^7}{1-q}=0\\1+2C_m=0\quad\Longrightarrow\quad C_m=-\frac12\end{gathered}\)
6. 6When the exponent is divisible by seven Add the seven unit terms directly \(\displaystyle \begin{gathered}q=1\\1+2C_m=7\quad\Longrightarrow\quad C_m=3\end{gathered}\)

Root data and complete explanations

Find ordered exact root data, factor coefficients and sums. Full written factorisation, proof and student-produced diagrams remain ungraded