Real Roots by Degree Parity

Summary: Euler’s §34–§37 results on when a polynomial must have real roots, based on the parity of its degree and the sign of its constant term.

Sources: chapter2

Last updated: 2026-04-23

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Odd-degree polynomials have a real root (§34)

Let $Z=z^{2n+1}+\alpha z^{2n}+\beta z^{2n-1}+\cdots$. As $z\to \infty$ the leading term dominates so $Z\to \infty$, and as $z\to -\infty$ Euler obtains $Z\to -\infty$ (source: chapter2, §34). By the intermediate-value-property, $Z$ takes every intermediate value, including $0$. Hence $Z$ has at least one real linear factor $z-c$.

Odd number of real linear factors in odd degree (§35)

If $Z$ has degree $2n+1$ and has a real linear factor $z-c$, dividing by $(z-c)(z-d)$ — if $z-d$ is a second real factor — yields a polynomial of odd degree $2n-1$, which again has a real linear factor. By induction:

If $Z$ has more than one real linear factor, it will have three, or (since the same argument is valid) five or seven, etc. (source: chapter2, §35)

Consequently the number of complex linear factors is even, consistent with §30.

Even-degree polynomials have an even number of real factors (§36)

Suppose a degree-$2n$ polynomial had an odd number $2m+1$ of real linear factors. Dividing by their product gives a quotient of odd degree $2n-2m-1$, which by §34 must have another real linear factor. So the count of real linear factors must be even (source: chapter2, §36).

Even degree with negative constant term (§37)

For $Z=z^{2n}\pm a{z}^{2n-1}\pm \cdots \pm nz-A$ with $A>0$:

• $Z\to +\infty$ as $z\to +\infty$,
• $Z=-A<0$ at $z=0$,
• $Z\to +\infty$ as $z\to -\infty$.

By the intermediate-value-property, $Z$ has a real root $c$ with $0<c<\infty$ and another real root $d$ with $-\infty <d<0$ (source: chapter2, §37).

Thus the equation $z^4+\alpha z^3+\beta z^2+\gamma z-a^2=0$ has two real roots, one positive and the other negative.

Euler’s use of infinity

Euler treats "$z=\infty$" as a legitimate input: he says $Z-\infty$ has linear factor $z-\infty$ and applies the intermediate-value-property on the interval $(-\infty ,+\infty )$. The reasoning is informal by modern standards but captures the right idea — the polynomial’s sign change at infinity forces a real root.

Related pages

• intermediate-value-property
• factoring-polynomials
• complex-conjugate-factors
• chapter-2-on-the-transformation-of-functions