If polynomials of degree at most $d$ converge at $d+1$ distinct points, their coefficients converge, by invertibility of the Vandermonde matrix. Their limit is a polynomial of degree at most $d$, and convergence is uniform on every compact subset of $\mathbb{C}$. Under convergence on compact sets, Rouché's theorem further locates a unique nearby zero for each simple root of the limit, for all sufficiently large indices. Finite initial polynomials need not have any root.

Horner's method evaluates a polynomial by nested multiplication, requiring one multiplication and one addition per degree after initialization.

For $p(X)=a_n{X}^n+\cdots +a_0$, start with $b_n=a_n$ and compute $b_j=x{b}_{j+1}+a_j$ for $j=n-1,\dots ,0$. The final value is $b_0=p(x)$. Equivalently,This reorganization reuses intermediate results, avoiding separate computation of every power. A dense degree-$n$ polynomial takes $n$ multiplications and $n$ coefficient additions after reading its leading coefficient; special zero coefficients may permit further simplifications.

For our cubic, the coefficient list in descending order is $1,-3,0,4$. The zero is important: it occupies the missing $X$ term's position. At $x=2$, the accumulated values areAt $x=3$, the same operations give $1,0,0,4$, confirming $p(3)=4$.

The intermediate values do more than evaluate. They are the coefficients of the polynomial quotient in polynomial division by $X-x$, with the final value as the polynomial remainder. At $x=2$, the first three values give $X^2-X-2$, soAt $x=3$, they give $X^2$, so $p(X)=(X-3)X^2+4$. This links an efficient numerical calculation with the exact factor theorem.

William George Horner's 1819 paper concerns solving numerical equations by successive approximation. The Royal Society archive records the manuscript and its reading on 1 July 1819; it provides a historical reference for the method's name. ^[ref]

// Target: geometry-and-topology.bigb

The reciprocal polynomial reverses the coefficients of a polynomial with nonzero constant term. For a binary cyclic code with generator $g$ and canonical check polynomial $h$, the reciprocal $h^{\ast }$ generates the dual of a cyclic code.

// Target: algebra.bigb

A quartic is a polynomial of degree of a polynomial four. Its leading coefficient is nonzero.

The Hasse derivatives of a polynomial are the coefficients in $f(x+t)={\sum }_{\ell \ge 0}{f}^{[\ell ]}(x)t^{\ell }$. A point $a$ has multiplicity at least $m$ exactly when $f^{[\ell ]}(a)=0$ for $0\le \ell <m$. Unlike iterated formal derivatives divided by factorials, this definition works in positive characteristic, where factorials can vanish. For example, in characteristic two $(x+1{)}^2$ has zero first formal derivative but nonzero second Hasse derivative.

// Target: coding-theory.bigb

Over a field of characteristic $p$, additive polynomials are sums of $p$-power monomials, $L(X)={\sum }_i{a}_i{X}^{p^i}$. They act linearly over the prime field. The identity follows from the Frobenius endomorphism; the converse follows by comparing mixed binomial coefficients. Their zero sets are additive groups, and a nonzero linear coefficient makes all roots simple.

An indeterminate is a formal symbol used to construct a polynomial. Choosing an input later defines a polynomial function; the symbol itself is not a fixed member of the coefficient ring.

Polynomial evaluation replaces the indeterminate of a polynomial by an input value. For a fixed scalar $a$, the rule $E_a:p\mapsto p(a)$ is a linear functional on a vector space of polynomials, because evaluating a linear combination gives the same linear combination of the values. For $p(t)=1+2t+3t^2$, evaluation at zero gives one and evaluation at two gives seventeen. Polynomial interpolation uses these measurements to reconstruct a polynomial subject to a polynomial degree bound.

A polynomial factorization expresses a polynomial as a product of other polynomials. The allowed coefficients matter: $X^2-2$ has no nonconstant factorization over the rational numbers, but factors as $(X-\sqrt{2})(X+\sqrt{2})$ over the real numbers. Difference of squares gives $X^2-Y^2=(X-Y)(X+Y)$ over any commutative ring.

An angiosperm pollen grain is a small male gametophyte, not itself a seed or a single gamete. Its tube-forming biological cell supports delivery of sperm.^[ref]

For $\dot{x}=x^2-\mu$, $\dot{y}=-\lambda y$, use entry $(x,h)$ and exit $(h,Y)$. If $\mu =k^2>0$, $x>k$, integration gives $Y=h[(h+k)(x-k)/((h-k)(x+k)){]}^{\lambda /(2k)}$. A smooth global return has $P(x)=\nu +cY+O(Y^2)$. If $\lambda >2k$, its derivative vanishes as $x\downarrow k$, and a small $\nu -k>0$ yields an attracting fixed point. If $\mu =-k^2<0$, the passage is $Y=h\exp [-(\lambda /k)(\arctan (h/k)-\arctan (x/k))]$. For small parameters the resulting global return is strongly contracting, with an attracting fixed point near $\nu$ on either side of zero.

For $\dot{x}=y+ax-b{x}^3$, $\dot{y}=x^3-x$, the function $H=y^2/2+x^2/2-x^4/4$ obeys $\dot{H}=(x-x^3)(ax-b{x}^3)$. Every nonconstant periodic orbit lies in $|x|<1$: at a maximum above one or a minimum below minus one, $\ddot{x}=x^3-x$ has the wrong sign; equality would force the solution to be an equilibrium by uniqueness. If $a\ne 0$ and $b/a<1$, then $\dot{H}=a{x}^2(1-x^2)(1-(b/a)x^2)$ has the strict sign of $a$ except at $x=0$, preventing a nonconstant periodic orbit.

Perfect plasticity idealizes a material whose yield surface does not evolve as plastic deformation accumulates. In one-dimensional elastic-perfectly plastic loading, $\sigma =E(\epsilon -{\epsilon }^p)$ while $|\sigma |\le {\sigma }_y$; continued plastic extension can occur at the fixed yield stress ${\sigma }_y$. Unloading can still be elastic, so perfect plasticity does not mean that the elastic response is absent.

This model isolates permanent flow from work hardening and is useful in limiting-load calculations. A rigid-perfectly plastic model makes the additional approximation of neglecting elastic strain, as in classical slip-line field theory. A real specimen may harden, soften, damage or show rate dependence, so the constant threshold is an idealization rather than a universal property of metals. It is also a local material law: decreasing cross-sectional area can change the measured axial force even when the local flow stress is constant.

Plastic strain rate is normal to the yield surface in the work-conjugate stress variables. With symmetric stress variations, the pairing is $D:\delta \sigma$; in two dimensions its off-diagonal contribution is $2D_{12}\delta {\sigma }_{12}$. This factor is essential when changing to reduced stress coordinates.

The Von Mises yield criterion depends only on the second invariant of deviatoric stress. With shear yield stress $k$, its surface is ${\sigma }^{\prime }:{\sigma }^{\prime }=2k^2$; the uniaxial yield stress is $\sqrt{3}k$. An associated flow rule gives plastic strain rate proportional to ${\sigma }^{\prime }$, up to a positive normalization of the plastic multiplier.

A control disturbance is an influence on the controlled plant that is not the chosen control input. In $\dot{x}=u+w$, a constant $w=0.2$ increases $x$ by $0.2$ per unit $t$ whenever $u=0$.

A plant output is a quantity exposed by the controlled plant, often modeled as $y=h(x,u,t)$. In the illustrative model it is $y=x$. An output of a computation has a different meaning.

A plant input is a signal supplied to the controlled plant. The chosen plant input is its control input; other influences can enter as control disturbances. This use concerns a dynamical system, rather than the input of a computation.

A seed packages a plant embryo, a seed food reserve and a protective seed coat. It is a living starting point for a new plant, not merely stored food. In a mature pea seed, the large cotyledons hold reserves; other seeds retain substantial endosperm. These are alternative arrangements, not successive germination stages.^[ref]

A useful reading exercise is to identify both the future plant and its initial food supply in each drawing. The large yellow area has the same role in both pictures, but it belongs to different plant tissues. Thus location and function alone do not establish tissue identity.

A flower is an angiosperm's reproductive structure. Stamens produce pollen; carpels enclose ovules. Petals and sepals surround these organs in a complete flower, but some flowers lack one or more groups, including either reproductive group.^[ref]

Read the drawing from outside inward, then trace two routes: where pollen is released, and where an ovule is enclosed. This reveals why catching pollen and forming a seed require different structures. A detached petal cannot substitute for a stigma simply because both belong to the same flower.