Mathematics · Ch 3 — Theory of Equations

Roots of Higher Degree Polynomial Equations

3.6

Roots of Higher Degree Polynomial Equations

Even without an exact formula, a handful of general facts help locate the real roots of a higher-degree polynomial equation $P(x)=0$ :

Every one-variable polynomial is a continuous function $\mathbb R\to\mathbb R$ , and is differentiable any number of times.
For an even-degree equation, $P(x)\to+\infty$ (assuming positive leading coefficient) as $x\to\pm\infty$ — the graph starts high on the left and ends high on the right (or the reverse, for a negative leading coefficient).
Every graphing fact from the Class 11 "graphing functions" toolkit still applies — e.g. changing only the constant term of $P$ shifts its graph purely up or down.
The real roots of $P(x)=0$ are exactly the $x$ -intercepts of the graph $y=P(x)$ .

Note

The Intermediate Value Property, applied to root-hunting. If $a<b$ are real numbers with $P(a)$ and $P(b)$ of opposite sign, then some $c$ between $a$ and $b$ satisfies $P(c)=0$ — i.e. there is a root strictly between $a$ and $b$ . It is not necessarily just one root in that interval — there could be $3,5,7,\ldots$ — but the number of real roots between $a$ and $b$ is always odd, never even, whenever $P(a)$ and $P(b)$ have opposite signs. …

Start your 14-day free trial to unlock the full solution →

← Previous · 3.5Q5 — Prove that a straight line and parabola cannot intersect at more than …Next · 3.7 →Polynomials with Additional Information