Chapter 9: On Trinomial Factors

Summary: Euler develops a constructive method for finding the real quadratic (“trinomial”) factors of any polynomial, then extends the method from polynomials to power series. Applied to $e^z$, $\sin z$, $\cos z$, the technique yields the first infinite-product representations of analytic functions: the sine product, the cosine product, and the exponential family. These products drive Chapter 10’s Basel solution and the family of even-zeta values.

Sources: chapter9

Last updated: 2026-04-29

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Overview

Chapter 2 promised that every real polynomial decomposes into real linear and real quadratic factors (the fundamental-theorem-of-algebra) but gave no algorithm. Chapter 9 supplies one — and then notices that the algorithm survives a passage to “polynomials of infinite degree”, producing a new representation theory for transcendental functions.

The chapter has three movements:

1. §143–§154 — the trinomial factorization theorem. Find the real quadratic factors of a polynomial directly, in the canonical form $p^2-2pqz\cos \varphi +q^2{z}^2$ (a trinomial factor). Using De Moivre, the factor condition splits into two real equations in the unknowns $r=p/q$ and $\varphi$. Applied to $a^n+z^n$ and $a^n-z^n$, the method produces the closed-form cyclotomic factorization indexed by equally spaced cosines (see factorization-of-an-plus-minus-zn).
2. §155–§158 — infinite products for the elementary transcendentals. Treat $e^z=(1+z/j{)}^j$ as a degree-$j$ polynomial with $j$ infinite, apply the Step-1 formulas, and pass to the limit. The result: infinite-product expansions for $e^x-1$, $(e^x-e^{-x})/2$, $(e^x+e^{-x})/2$ (see exponential-infinite-product). Substituting $x=iz$ via eulers-formula gives the sine and cosine infinite products.
3. §159–§164 — generalization to $e^z-2\cos g+e^{-z}$ and circular reformulations. The same recipe applied to a parametrized “binomial-of-binomials” produces a family of infinite products for $(\cos v+\cos g)/(1+\cos g)$, $(\cos v-\cos g)/(1-\cos g)$, $(\sin g+\sin v)/\sin g$, and a fourth variant — collectively the four formulas of §163, plus their circular-arc reformulations in §164.

See also: trinomial-factor, factorization-of-an-plus-minus-zn, exponential-infinite-product, sine-infinite-product, cosine-infinite-product.

Structure of the chapter

§143–§144 — Why trinomial factors

A linear factor $p-qz$ of a polynomial $\alpha +\beta z+\gamma z^2+\cdots$ corresponds to a root $z=p/q$. Real linear factors are easy in principle (solve the equation); the difficulty is the complex roots. Since complex roots come in conjugate pairs (see complex-conjugate-factors), their product is a real quadratic. Euler’s strategy: search for the real quadratic factor directly, in a parametrization that exposes the discriminant condition.

§145 — The canonical form

A real quadratic $p-qz+r{z}^2$ has complex linear factors when $4pr>q^2$, i.e. $q/(2\sqrt{pr})<1$. Set $q/(2\sqrt{pr})=\cos \varphi$ for some real arc $\varphi$, absorb $\sqrt{pr}$ into the names $p,q$, and the factor takes the canonical form

$p^2-2pqz\cos \varphi +q^2{z}^2,$

with complex linear factors $qz-p(\cos \varphi \pm i\sin \varphi )$. When $\cos \varphi =\pm 1$ the two linear factors coincide and are real. See trinomial-factor.

§146–§149 — Two real equations for $r$ and $\varphi$

If the canonical trinomial divides a polynomial $\alpha +\beta z+\gamma z^2+\cdots$, then the polynomial vanishes at $z=(p/q)(\cos \varphi \pm i\sin \varphi )$. Substitute, use de-moivre-formula $z^n=r^n(\cos n\varphi \pm i\sin n\varphi )$ where $r=p/q$, separate real and imaginary parts, and obtain

$0=\alpha +\beta r\cos \varphi +\gamma r^2\cos 2\varphi +\delta r^3\cos 3\varphi +\cdots ,$

$0=\beta r\sin \varphi +\gamma r^2\sin 2\varphi +\delta r^3\sin 3\varphi +\cdots .$

(source: chapter9, §148). Two real equations, two unknowns. §149 gives a slick generalization: multiplying by $\cos m\varphi$ and $\sin m\varphi$ before adding produces the family

$0=\alpha \cos m\varphi +\beta r\cos (m\mp 1)\varphi +\gamma r^2\cos (m\mp 2)\varphi +\cdots ,$

any pair of which determines $r$ and $\varphi$. Multiple solutions yield multiple trinomial factors, and (Euler claims) all of them.

§150 — Factoring $a^n+z^n$

Apply the §148 method with $\alpha =a^n$ and only $\zeta z^n$ nonzero. The second equation $r^n\sin n\varphi =0$ gives $n\varphi =k\pi$, the first then forces $\cos n\varphi =-1$, so $n\varphi =(2k+1)\pi$ and $r=a$. Trinomial factor:

$a^2-2az\cos \frac{(2k+1)\pi }{n}+z^2.$

Run $2k+1$ over odd integers $\le n$. If $n$ is odd, $2k+1=n$ produces the perfect square $(a+z{)}^2$, giving the lone real linear factor $a+z$. See factorization-of-an-plus-minus-zn.

§151 — Factoring $a^n-z^n$

The same setup, but the first equation now requires $\cos n\varphi =+1$, hence $n\varphi =2k\pi$. Trinomial factor:

$a^2-2az\cos \frac{2k\pi }{n}+z^2.$

$2k=0$ gives $(a-z{)}^2$ → real linear factor $a-z$; if $n$ is even, $2k=n$ gives $(a+z{)}^2$ → real linear factor $a+z$. See factorization-of-an-plus-minus-zn.

§152–§153 — The master formula

The expression $a^{2n}-2a^n{z}^n\cos g+z^{2n}$ has no factor of the form $\eta +\theta z^n$ but is the product of two complex factors $a^n-z^n{e}^{\pm ig}$. Apply §149 with $m=2n$ to get $r=a$, $\cos n\varphi =\cos g$, hence $n\varphi =2k\pi \pm g$. Trinomial factor:

$a^2-2az\cos \frac{2k\pi \pm g}{n}+z^2.$

This subsumes §150 ($g=\pi$, after factoring out a square root) and §151 ($g=0$). See factorization-of-an-plus-minus-zn.

§154 — Every polynomial admits real factorization

§154 stitches the pieces together. A polynomial $\alpha +\beta z^n+\gamma z^{2n}$ either has two real factors of the form $\eta +\theta z^n$ (each handled by §150–§151) or has none, in which case it is exactly an instance of §153. Polynomials of higher degree in $z^n$ reduce in the same way. Conclusion: “every polynomial can be expressed as a product of real linear and real quadratic factors” — the existence claim of §32 is now constructive, modulo finding the cosine arcs.

§155 — Infinite product for $e^z-1$

From Chapter 7, $e^z-1=(1+z/j{)}^j-1$ for $j$ infinite. Compare with §151 ($a^n-z^n$ at $a=1+z/j$, $z=1$, $n=j$). Each factor $a^2-2az\cos (2k\pi /j)+z^2$ becomes, after using $\cos (2k\pi /j)=1-2k^2{\pi }^2/j^2$ from §134 and dropping high-order $j$ terms,

$\propto 1+\frac{z}{j}+\frac{z^2}{4k^2{\pi }^2}.$

The $k=0$ factor degenerates to $z^2/j^2$; take the square root $z/j\propto z$. Hence $e^z-1=z{\prod }_{k=1}^{\infty }(1+z/j+z^2/(4k^2{\pi }^2))$ — but the $z/j$ terms are infinitesimal yet collectively non-negligible (there are $\frac{1}{2}j$ of them, contributing $z/2$). Euler resolves this in §156.

§156 — Infinite product for $(e^x-e^{-x})/2$

Now compare $(1+x/j{)}^j-(1-x/j{)}^j$ (which by binomial is $2(x+x^3/3!+x^5/5!+\cdots )=e^x-e^{-x}$) with §151 at $a=1+x/j$, $z=1-x/j$, $n=j$. Each factor cleans up to $1+x^2/(k^2{\pi }^2)$ (no troublesome $x/j$ remainder), and the $k=0$ factor square-roots to $x$:

$\frac{e^x-e^{-x}}{2}=x{\prod }_{k=1}^{\infty }\left(1+\frac{x^2}{k^2{\pi }^2}\right).$

See exponential-infinite-product.

§157 — Infinite product for $(e^x+e^{-x})/2$

Same setup with §150 ($a^n+z^n$): each factor has arc $(2k+1)\pi /j$, and after simplification,

$\frac{e^x+e^{-x}}{2}={\prod }_{k=0}^{\infty }\left(1+\frac{4x^2}{(2k+1{)}^2{\pi }^2}\right).$

See exponential-infinite-product.

§158 — Sine and cosine infinite products

Substitute $x=iz$ into §156 and §157, using eulers-formula $(e^{iz}-e^{-iz})/(2i)=\sin z$ and $(e^{iz}+e^{-iz})/2=\cos z$:

$\sin z=z{\prod }_{k=1}^{\infty }\left(1-\frac{z^2}{k^2{\pi }^2}\right),$

$\cos z={\prod }_{k=0}^{\infty }\left(1-\frac{4z^2}{(2k+1{)}^2{\pi }^2}\right).$

The zeros of $\sin z$ ($z=k\pi$) and $\cos z$ ($z=(2k+1)\pi /2$) appear directly as the roots of the respective factors. See sine-infinite-product and cosine-infinite-product.

§159–§163 — The $e^z-2\cos g+e^{-z}$ family

Apply §152 to $(1+x/j{)}^j-2\cos g+(1-x/j{)}^j$ with $2n=j$, after which the factor takes the form $(2k\pi \pm g)$ in the cosine. Cleaning up:

$\frac{e^x-2\cos g+e^{-x}}{2(1-\cos g)}={\prod }_{k=1}^{\infty }\left(1+\frac{x^2}{(2k\pi -g{)}^2}\right)\left(1+\frac{x^2}{(2k\pi +g{)}^2}\right)\cdot \left(1+\frac{x^2}{g^2}\right).$

(source: chapter9, §159, with the convention that $k$ runs from $0$ for the trailing factor). Substituting $zi$ for $x$ converts this into

$\frac{\cos z-\cos g}{1-\cos g}=\left(1-\frac{z}{g}\right)\left(1+\frac{z}{g}\right)\left(1-\frac{z}{2\pi -g}\right)\left(1+\frac{z}{2\pi -g}\right)\cdots ,$

which, in the limit $g\to 0$, recovers the cosine-infinite-product (after some bookkeeping). §160–§162 specialize the parameters; §163 collects four formulas:

$\frac{\cos v+\cos g}{1+\cos g},\frac{\cos v-\cos g}{1-\cos g},\frac{\sin g+\sin v}{\sin g},\frac{\sin g-\sin v}{\sin g}$

each as an explicit infinite product in $v$ (with $g$ a parameter). The systematic catalog establishes that the trinomial-factor technique is general — every elementary trig combination has an infinite product expansion.

§164 — Reformulation using arcs

The same expressions written in terms of $\cos z+\tan (g/2)\sin z$, $\cos (z)-\cot (g/2)\sin z$, etc. The point is that if $b$ and $c$ in the §161 formulas are interpreted as $b=0$, $c=ig$, then $e^c\pm e^{-c}=2\cos g$, $\pm 2i\sin g$ — and the abstract identities of §161 become circular-trigonometric statements (source: chapter9, §164). Euler closes: “The law of formation for these factors is sufficiently simple and uniform.”

Notable points

• The trinomial form encodes polar coordinates. $p^2-2pqz\cos \varphi +q^2{z}^2$ is exactly $(qz-p{e}^{i\varphi })(qz-p{e}^{-i\varphi })$, with $p/q$ playing the role of the modulus and $\varphi$ the argument of the complex root. Euler does not invoke $e^{i\varphi }$ here, but the polar form is implicit.
• The §148 system is De Moivre in disguise. The two real equations are the real and imaginary parts of a single complex equation $\sum c_n{r}^n{e}^{in\varphi }=0$. Euler is doing complex analysis without writing complex exponentials, decades before the conceptual machinery existed.
• The cyclotomic factorization predates the term. §150–§151 are the modern factorizations of $z^n-a^n$ over $\mathbb{R}$, indexed by primitive roots of unity. Gauss’s Disquisitiones (1801), where the term cyclotomic arises, is half a century later.
• Infinite products are the dual representation to power series. A power series captures local behavior at $0$; an infinite product captures global zero structure. Euler is the first to systematically exploit both.
• The sine product is the precursor of Weierstrass factorization. Modern complex analysis recovers Euler’s formula as the canonical product representation of an entire function of order 1, with explicit elementary factors and a genus-1 exponential. Euler’s manipulation, free of any convergence theory, hits the right answer because $\sin z/z$ has order exactly $1$ and no genus-1 correction is needed.
• The Basel problem is one comparison away. Equating $z^3$ coefficients in $\sin z=z-z^3/6+\cdots$ and $\sin z=z\prod (1-z^2/k^2{\pi }^2)$ gives $\sum 1/k^2={\pi }^2/6$. Euler does the comparison in Chapter 10 (see basel-problem).

Why this chapter matters

Chapter 9 is where Euler turns factorization — a tool for polynomials — into a representation theory for transcendentals. The constructive solution of the §32 problem is satisfying on its own, but the real prize is the second movement: $\sin z$, $\cos z$, $e^x\pm e^{-x}$ all become products. Each product is a list of zeros; each zero is a piece of geometric or algebraic data. Comparing product expansions to power-series expansions (Chapter 10) instantly evaluates infinite sums that resisted Bernoulli, Leibniz, and the entire mathematical world for a generation.

The technique extends much further than Chapter 9 demonstrates. Euler will use the same products to construct the partial-fraction expansions of $\cot z$, $\tan z$, $\csc z$, and to derive the entire family of even-zeta values $\zeta (2),\zeta (4),\zeta (6),\dots$. The shape of nineteenth-century complex analysis — Mittag-Leffler, Hadamard, Weierstrass — is laid out here in embryo.

Related pages

• trinomial-factor
• factorization-of-an-plus-minus-zn
• exponential-infinite-product
• sine-infinite-product
• cosine-infinite-product
• de-moivre-formula
• eulers-formula
• sine-and-cosine-series
• exponential-series
• factoring-polynomials
• complex-conjugate-factors
• fundamental-theorem-of-algebra
• chapter-2-on-the-transformation-of-functions
• chapter-7-on-exponentials-and-logarithms-expressed-through-series
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle
• chapter-10-on-the-use-of-the-discovered-factors-to-sum-infinite-series
• basel-problem
• zeta-at-even-integers
• newtons-identities
• circular-arc-series
• cotangent-partial-fraction