Conjugate roots and De Moivre’s theorem

If a polynomial ($p(x)$) has real coefficients and $a+bi$ is a non-real root, then (a-bi) is also a root. This is the conjugate-root theorem.

The reason is that conjugation commutes with addition and multiplication. For real coefficients, $p(\overline z)=\overline{p(z)}$. If $p(z)=0$, then $p(\overline z)=\overline0=0$.

Non-real roots therefore occur in pairs when the coefficients are real. A real polynomial of odd degree must still have at least one real root because the remaining non-real roots pair off.

The theorem does not apply automatically to polynomials with non-real coefficients.

The conjugate roots $a\pm bi$ produce the real quadratic factor $(x-a-bi)(x-a+bi)=(x-a)^2+b^2$.

Expanded, this factor is $x^2-2ax+(a^2+b^2)$. Its discriminant is $-4b^2<0$ when $b\ne0$, confirming that its roots are non-real.

To construct a real polynomial from known roots, include every required conjugate, write one factor per root, combine conjugate factors first, then expand only if requested.

Multiplicity is preserved: if $a+bi$ is a repeated root of multiplicity m in a real polynomial, (a-bi) has the same multiplicity.

De Moivre's theorem states that for an integer n, raising a polar complex number to the nth power raises its modulus to n and multiplies its argument by n.

For $z=r\operatorname{cis}\theta$, $z^n=r^n\operatorname{cis}(n\theta)$. In full trigonometric form, $(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)$.

For positive integers, the result follows by repeated polar multiplication. For negative integers it also holds when $z\ne0$, using reciprocals.

The theorem turns a long Cartesian expansion into two simple operations, but the original number must first be converted to polar form accurately.

To calculate a large power, convert the base to polar form, apply De Moivre's theorem, normalize the angle if useful, and convert back to Cartesian form only if requested.

Exact standard angles make the final conversion exact. Keep radical moduli such as $\sqrt2$ symbolic before raising them to a power.

A reasonableness check is immediate: $|z^n|=|z|^n$. If the computed Cartesian result has a different modulus, revisit the arithmetic.

When the multiplied argument completes whole turns, the power may become real even if the original number is non-real.

De Moivre's theorem can generate trigonometric identities by expanding $(\cos\theta+i\sin\theta)^n$ algebraically and comparing real and imaginary parts with $\cos(n\theta)+i\sin(n\theta)$.

The real parts must be equal, and the imaginary parts must be equal. This produces exact formulas for $\cos(n\theta)$ and $\sin(n\theta)$.

For $n = 3$, expansion gives the triple-angle identities. Equivalent forms can be obtained using $\sin^2\theta=1-\cos^2\theta$ or $\cos^2\theta=1-\sin^2\theta$.

Show the comparison step explicitly; a final identity without the complex expansion may omit the central reasoning.

To solve $z^n=w$, write $w=R\operatorname{cis}\theta$ with $R>0$, and write $z=\rho\operatorname{cis}\phi$. De Moivre gives $z^n=\rho^n\operatorname{cis}(n\phi)$.

Equating moduli gives $\rho=R^{\frac{1}{n}}$. Equating directions requires $n\phi=\theta+2k\pi$, so $\phi=\frac{\theta+2k\pi}{n}$.

Exactly n distinct roots are obtained by using $k=0,1,\ldots,n-1$. Other integer values of k repeat these roots.

The roots all have the same modulus $R^{\frac{1}{n}}$ and successive arguments differ by $\frac{2\pi}{n}$.

Use the nth-root formula systematically: express the target in polar form, find the root modulus, create the n arguments, and convert roots to Cartesian form only if required.

A complete answer should list all distinct roots. Order them by increasing argument or by increasing k so omissions are easy to detect.

Check one root by raising it to the nth power. Check the whole set geometrically: equal modulus and equal angular spacing are required.

When the target is a negative real number, use argument $\pi$ under the principal convention before generating all roots.

The n roots of a non-zero complex number lie on the circle centered at the origin with radius $R^{\frac{1}{n}}$. Their arguments are equally spaced by $\frac{2\pi}{n}$.

Joining consecutive roots forms a regular n-gon. Rotating one root by $\frac{2\pi}{n}$ reaches the next root.

Changing the argument θ of the target rotates the entire regular polygon by $\frac{\theta}{n}$, while changing the modulus R changes only its radius to $R^{\frac{1}{n}}$.

The sum of all n roots is zero for $n>1$, because the symmetric position vectors cancel. This also follows from the coefficient of $z^{n-1}$ in $z^n-w$.

The nth roots of unity solve $z^n=1$. Since $1=\operatorname{cis}(2m\pi)$, the distinct roots are $\omega_k=\operatorname{cis}\frac{2k\pi}{n}$ for $k=0,1,\ldots,n-1$.

All roots of unity have modulus 1 and form a regular n-gon on the unit circle. The root $1$ always occurs at angle 0.

The sixth roots lie on the unit circle with angular separation $\frac{\pi}{3}$, forming a regular hexagon.

The visual shows the sixth roots of unity: six points on one circle, separated by $\frac{2\pi}{6}=\frac{\pi}{3}$, forming a regular hexagon.

A primitive nth root of unity generates every nth root by repeated powers. The root $\operatorname{cis}\frac{2k\pi}{n}$ is primitive exactly when k and n are coprime.

Polynomial equations can be solved by combining factorization, conjugate pairing, and nth-root methods. First identify the algebraic structure before choosing a technique.

For $z^n=w$, use the nth-root formula directly. For a real polynomial with one known non-real root, add its conjugate and divide by the resulting real quadratic factor.

After reducing the polynomial degree, solve the remaining real or complex factor by familiar methods. Count roots with multiplicity to confirm completeness.

The fundamental theorem of algebra guarantees exactly n complex roots counting multiplicity for a degree-n polynomial.

Advanced questions may combine a power, a root condition, and a polynomial requirement. Keep the modulus and argument work separate until the final form is requested.

When an equation can be rearranged to $z^n=w$, use the complete root family rather than taking only a principal real-style root.

Symmetry can simplify expressions involving roots of unity. For example, $1+\omega+\omega^2=0$ for the non-real cube roots of unity.

Use exact polar values first; convert to Cartesian form only for factorization, addition, or a requested final answer.

Use complex mode and radian angle mode. A GDC can evaluate powers, convert between Cartesian and polar form, and numerically solve many polynomial equations.

For nth roots, a calculator may display only one principal root. The mathematical formula with $k=0,1,\ldots,n-1$ is still required to generate the complete set.

When a polynomial solver returns decimals, identify exact conjugate structure where possible. Values such as $1.73205$ may represent $\sqrt3$, but claim an exact value only when algebra supports it.

Use the GDC to verify each proposed root by evaluating the original expression. Small numerical residues such as $10^{-12}$ are rounding effects, not additional roots.

For an Argand sketch, plot all roots and check equal radii and angular gaps. The visual check complements the numerical check.

The most frequent errors are forgetting a conjugate root, applying the conjugate-root theorem without real coefficients, using $nr$ instead of $r^n$, dividing only the principal argument by n, or listing fewer than n roots.

For powers, show the polar conversion and both De Moivre operations. For roots, show the target polar form, the common root modulus, the indexed angle formula, and the complete k range.

For polynomial construction, pair conjugates before expanding and count the degree. For an Argand answer, state the radius, angular separation, and regular polygon.

Normalize arguments only when the question requests a principal interval; coterminal root angles are mathematically valid but should be presented consistently.

Attempt each problem before reading the steps. The questions progress from conjugate-root fundamentals to powers, complete root families, roots of unity and mixed polynomial reasoning.

Use exact values and radian arguments unless a question explicitly requests a decimal approximation.

Conjugate pairing connects non-real roots to real polynomial factors. De Moivre's theorem then makes powers and complete root families systematic in polar form.

A strong final check combines algebraic substitution, root count, common modulus, equal angular spacing and the polynomial's coefficient conditions.