§1.11 Zeros of Polynomials

Keywords:: polynomials, zeros of polynomials
Referenced by:: §1.1
Permalink:: http://dlmf.nist.gov/1.11

§1.11(i) Division Algorithm
§1.11(ii) Elementary Properties
§1.11(iii) Polynomials of Degrees Two, Three, and Four
§1.11(iv) Roots of Unity and of Other Constants
§1.11(v) Stable Polynomials

§1.11(i) Division Algorithm

Notes:: For the Horner scheme, see Burnside and Panton (1960, pp. 8–9). The double Horner scheme is derived similarly. For (1.11.7), see Dummit and Foote (1999, pp. 300–301).
Keywords:: zeros of polynomials
Referenced by:: §18.40, §3.8(iv)
Permalink:: http://dlmf.nist.gov/1.11.i

¶ Horner’s Scheme

Keywords:: Horner’s scheme for polynomials, zeros of polynomials

Let

1.11.1 $f(z)=a_{n}z^{n}+a_{{n-1}}z^{{n-1}}+\dots+a_{0}.$

Symbols:: $z$ : variable and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §1.11(i)
Permalink:: http://dlmf.nist.gov/1.11.E1
Encodings:: TeX, pMML, png

Then

1.11.2 $f(z)=(z-\alpha)(b_{n}z^{{n-1}}+b_{{n-1}}z^{{n-2}}+\dots+b_{1})+b_{0},$

Symbols:: $z$ : variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.11.E2
Encodings:: TeX, pMML, png

where $b_{n}=a_{n}$ ,

1.11.3 $b_{k}=\alpha b_{{k+1}}+a_{k},$ $k=n-1,n-2,\dots,0$ ,

Symbols:: $k$ : integer and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §1.11(i)
Permalink:: http://dlmf.nist.gov/1.11.E3
Encodings:: TeX, pMML, png

1.11.4 $f(\alpha)=b_{0}.$

Permalink:: http://dlmf.nist.gov/1.11.E4
Encodings:: TeX, pMML, png

¶ Extended Horner Scheme

Keywords:: Horner’s scheme for polynomials, zeros of polynomials

With $b_{k}$ as in (1.11.1)–(1.11.3) let $c_{n}=a_{n}$ and

1.11.5 $c_{k}=\alpha c_{{k+1}}+b_{k},$ $k=n-1,n-2,\dots,1$ .

Symbols:: $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.11.E5
Encodings:: TeX, pMML, png

Then

1.11.6 $f^{{\prime}}(\alpha)=c_{1}.$

Permalink:: http://dlmf.nist.gov/1.11.E6
Encodings:: TeX, pMML, png

More generally, for polynomials $f(z)$ and $g(z)$ , there are polynomials $q(z)$ and $r(z)$ , found by equating coefficients, such that

1.11.7 $f(z)=g(z)q(z)+r(z),$

Symbols:: $z$ : variable, $q(z)$ : polynomial and $r(z)$ : polynomial
Referenced by:: §1.11(i)
Permalink:: http://dlmf.nist.gov/1.11.E7
Encodings:: TeX, pMML, png

where $0\leq\deg r(z)<\deg g(z)$ .

§1.11(ii) Elementary Properties

Notes:: See Burnside and Panton (1960, Chapter 2). For (1.11.9) see Dummit and Foote (1999, p. 591).
Keywords:: monic polynomial, polynomials, zeros of polynomials
Referenced by:: §23.3(i)
Permalink:: http://dlmf.nist.gov/1.11.ii

A polynomial of degree $n$ with real or complex coefficients has exactly $n$ real or complex zeros counting multiplicity. Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term. A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative.

¶ Descartes’ Rule of Signs

Keywords:: Descartes’ rule of signs (for polynomials), zeros of polynomials

The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. A similar relation holds for the changes in sign of the coefficients of $f(-z)$ , and hence for the number of negative zeros of $f(z)$ .

¶ Example

Keywords:: discriminant, polynomials, zeros of polynomials

1.11.8

$f(z)=z^{8}+10z^{3}+z-4,$

$f(-z)=z^{8}-10z^{3}-z-4.$

Symbols:: $z$ : variable
Permalink:: http://dlmf.nist.gov/1.11.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros.

Next, let $f(z)=a_{n}z^{n}+a_{{n-1}}z^{{n-1}}+\dots+a_{0}$ . The zeros of $z^{n}f(1/z)=a_{0}z^{n}+a_{1}z^{{n-1}}+\dots+a_{n}$ are reciprocals of the zeros of $f(z)$ .

The discriminant of $f(z)$ is defined by

1.11.9 $D=a_{n}^{{2n-2}}\prod _{{j<k}}(z_{j}-z_{k})^{2},$

Symbols:: $z$ : variable, $j$ : integer, $k$ : integer, $n$ : nonnegative integer and $D$ : discriminant of $f(z)$
Referenced by:: §1.11(ii)
Permalink:: http://dlmf.nist.gov/1.11.E9
Encodings:: TeX, pMML, png

where $z_{1},z_{2},\dots,z_{n}$ are the zeros of $f(z)$ . The elementary symmetric functions of the zeros are (with $a_{n}\not=0$ )

1.11.10

$\displaystyle z_{1}+z_{2}+\dots+z_{n} =-a_{{n-1}}/a_{n},$

$\displaystyle \sum _{{1\leq j<k\leq n}}z_{j}z_{k} =a_{{n-2}}/a_{n},$

$\displaystyle \mathrel{\vdots}$

$\displaystyle z_{1}z_{2}\cdots z_{n} =(-1)^{n}a_{0}/a_{n}.$

Symbols:: $z$ : variable, $j$ : integer, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.11.E10
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

§1.11(iii) Polynomials of Degrees Two, Three, and Four

Notes:: See Dummit and Foote (1999, pp. 592–595, 611–616).
Keywords:: zeros of polynomials
Referenced by:: ¶ ‣ §23.22(ii), §3.8(iv), §4.43
Permalink:: http://dlmf.nist.gov/1.11.iii

¶ Quadratic Equations

Keywords:: quadratic equations

The roots of $az^{2}+bz+c=0$ are

1.11.11

$\frac{-b\pm\sqrt{D}}{2a},$

$D=b^{2}-4ac.$

Symbols:: $D$ : discriminant
Permalink:: http://dlmf.nist.gov/1.11.E11
Encodings:: TeX, TeX, pMML, pMML, png, png

The sum and product of the roots are respectively $-b/a$ and $c/a$ .

¶ Cubic Equations

Keywords:: cubic equation

Set $z=w-\tfrac{1}{3}a$ to reduce $f(z)=z^{3}+az^{2}+bz+c$ to $g(w)=w^{3}+pw+q$ , with $p=(3b-a^{2})/3$ , $q=(2a^{3}-9ab+27c)/27$ . The discriminant of $g(w)$ is

1.11.12 $D=-4p^{3}-27q^{2}.$

Symbols:: $p$ , $q$ and $D$ : discriminant
A&S Ref:: 3.8.1
Permalink:: http://dlmf.nist.gov/1.11.E12
Encodings:: TeX, pMML, png

Let

1.11.13

$A=\sqrt[3]{-\tfrac{27}{2}q+\tfrac{3}{2}\sqrt{-3D}},$

$B=-3p/A.$

Symbols:: $p$ , $q$ , $D$ : discriminant, $A$ and $B$
Permalink:: http://dlmf.nist.gov/1.11.E13
Encodings:: TeX, TeX, pMML, pMML, png, png

The roots of $g(w)=0$ are

1.11.14

$\tfrac{1}{3}(A+B),$

$\tfrac{1}{3}(\rho A+\rho^{2}B),$

$\tfrac{1}{3}(\rho^{2}A+\rho B),$

Symbols:: $\rho$ , $A$ and $B$
Permalink:: http://dlmf.nist.gov/1.11.E14
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

with

1.11.15

$\rho=-\tfrac{1}{2}+\tfrac{1}{2}\sqrt{-3}=e^{{2\pi i/3}},$

$\rho^{2}=e^{{-2\pi i/3}}.$

Symbols:: $e$ : base of exponential function and $\rho$
Permalink:: http://dlmf.nist.gov/1.11.E15
Encodings:: TeX, TeX, pMML, pMML, png, png

Addition of $-\frac{1}{3}a$ to each of these roots gives the roots of $f(z)=0$ .

¶ Example

$f(z)=z^{3}-6z^{2}+6z-2$ , $g(w)=w^{3}-6w-6$ , $A=3\sqrt[3]{4}$ , $B=3\sqrt[3]{2}$ . Roots of $f(z)=0$ are $2+\sqrt[3]{4}+\sqrt[3]{2}$ , $2+\sqrt[3]{4}\rho+\sqrt[3]{2}\rho^{2}$ , $2+\sqrt[3]{4}\rho^{2}+\sqrt[3]{2}\rho$ .

For another method see §4.43.

¶ Quartic Equations

Keywords:: cubic equation, quartic equations, resolvent cubic equation, zeros of polynomials

Set $z=w-\tfrac{1}{4}a$ to reduce $f(z)=z^{4}+az^{3}+bz^{2}+cz+d$ to

1.11.16

$g(w)=w^{4}+pw^{2}+qw+r,$

$p=(-3a^{2}+8b)/8,$

$q=(a^{3}-4ab+8c)/8,$

$r=(-3a^{4}+16a^{2}b-64ac+256d)/256.$

Symbols:: $w$ : variable, $p$ , $q$ and $r$
Permalink:: http://dlmf.nist.gov/1.11.E16
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

The discriminant of $g(w)$ is

1.11.17 $D=16p^{4}r-4p^{3}q^{2}-128p^{2}r^{2}+144pq^{2}r-27q^{4}+256r^{3}.$

Symbols:: $p$ , $q$ , $r$ and $D$ : discriminant
Permalink:: http://dlmf.nist.gov/1.11.E17
Encodings:: TeX, pMML, png

For the roots $\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4}$ of $g(w)=0$ and the roots $\theta _{1},\theta _{2},\theta _{3}$ of the resolvent cubic equation

1.11.18 $z^{3}-2pz^{2}+(p^{2}-4r)z+q^{2}=0,$

Symbols:: $z$ : variable, $p$ , $q$ and $r$
Permalink:: http://dlmf.nist.gov/1.11.E18
Encodings:: TeX, pMML, png

we have

1.11.19

$2\alpha _{1}=\phantom{+}\sqrt{-\theta _{1}}+\sqrt{-\theta _{2}}+\sqrt{-\theta _{3}},$

$2\alpha _{2}=\phantom{+}\sqrt{-\theta _{1}}-\sqrt{-\theta _{2}}-\sqrt{-\theta _{3}},$

$2\alpha _{3}=-\sqrt{-\theta _{1}}+\sqrt{-\theta _{2}}-\sqrt{-\theta _{3}},$

$2\alpha _{4}=-\sqrt{-\theta _{1}}-\sqrt{-\theta _{2}}+\sqrt{-\theta _{3}}.$

Symbols:: $j$ : integer and $\theta _{j}$ : cubic roots
Permalink:: http://dlmf.nist.gov/1.11.E19
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

The square roots are chosen so that

1.11.20 $\sqrt{-\theta _{1}}\;\sqrt{-\theta _{2}}\;\sqrt{-\theta _{3}}=-q.$

Symbols:: $q$ and $\theta _{j}$ : cubic roots
Permalink:: http://dlmf.nist.gov/1.11.E20
Encodings:: TeX, pMML, png

Add $-\tfrac{1}{4}a$ to the roots of $g(w)=0$ to get those of $f(z)=0$ .

¶ Example

$f(z)=z^{4}-4z^{3}+5z+2$ , $g(w)=w^{4}-6w^{2}-3w+4$ . Resolvent cubic is $z^{3}+12z^{2}+20z+9=0$ with roots $\theta _{1}=-1$ , $\theta _{2}=-\tfrac{1}{2}(11+\sqrt{85})$ , $\theta _{3}=-\tfrac{1}{2}(11-\sqrt{85})$ , and $\sqrt{-\theta _{1}}=1$ , $\sqrt{-\theta _{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})$ , $\sqrt{-\theta _{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})$ . So $2\alpha _{1}=1+\sqrt{17}$ , $2\alpha _{2}=1-\sqrt{17}$ , $2\alpha _{3}=-1+\sqrt{5}$ , $2\alpha _{4}=-1-\sqrt{5}$ , and the roots of $f(z)=0$ are $\tfrac{1}{2}(3\pm\sqrt{17})$ , $\tfrac{1}{2}(1\pm\sqrt{5})$ .

§1.11(iv) Roots of Unity and of Other Constants

Notes:: See Burnside and Panton (1960, pp. 80–81).
Keywords:: constants, unity, zeros of polynomials
Permalink:: http://dlmf.nist.gov/1.11.iv

The roots of

1.11.21 $z^{n}-1=(z-1)(z^{{n-1}}+z^{{n-2}}+\dots+z+1)=0$

Symbols:: $z$ : variable and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §1.3(ii)
Permalink:: http://dlmf.nist.gov/1.11.E21
Encodings:: TeX, pMML, png

are 1, $e^{{2\pi i/n}}$ , $e^{{4\pi i/n}},\dots,e^{{(2n-2)\pi i/n}}$ , and of $z^{n}+1=0$ they are $e^{{\pi i/n}},e^{{3\pi i/n}},\dots,e^{{(2n-1)\pi i/n}}$ .

The roots of

1.11.22 $z^{n}=a+ib,$ $a,b$ real,

Symbols:: $z$ : variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.11.E22
Encodings:: TeX, pMML, png

are

1.11.23 $\sqrt[n]{R}\left(\mathop{\cos\/}\nolimits\!\left(\frac{\alpha+2k\pi}{n}\right)+i\mathop{\sin\/}\nolimits\!\left(\frac{\alpha+2k\pi}{n}\right)\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : integer, $n$ : nonnegative integer and $R$ : radius
Permalink:: http://dlmf.nist.gov/1.11.E23
Encodings:: TeX, pMML, png

where $R=(a^{2}+b^{2})^{{1/2}}$ , $\alpha=\mathop{\mathrm{ph}\/}\nolimits\!\left(a+ib\right)$ , with the principal value of phase (§1.9(i)), and $k=0,1,\dots,n-1$ .

§1.11(v) Stable Polynomials

Notes:: See Henrici (1977, vol. 2, pp. 555–559).
Keywords:: polynomials, stable polynomials
Permalink:: http://dlmf.nist.gov/1.11.v

1.11.24 $f(z)=a_{0}+a_{1}z+\dots+a_{n}z^{n},$

Symbols:: $z$ : variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.11.E24
Encodings:: TeX, pMML, png

with real coefficients, is called stable if the real parts of all the zeros are strictly negative.

¶ Hurwitz Criterion

Keywords:: Hurwitz criterion for stable polynomials, stable polynomials

Let

1.11.25

$D_{1}=a_{1},$

$D_{2}=\begin{vmatrix}a_{1}&a_{3}\\ a_{0}&a_{2}\end{vmatrix},$

$D_{3}=\begin{vmatrix}a_{1}&a_{3}&a_{5}\\ a_{0}&a_{2}&a_{4}\\ 0&a_{1}&a_{3}\end{vmatrix},$

Symbols:: $D_{j}$ : quantities
Permalink:: http://dlmf.nist.gov/1.11.E25
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

and

1.11.26 $D_{k}=\det[h_{k}^{{(1)}},h_{k}^{{(3)}},\dots,h_{k}^{{(2k-1)}}],$

Symbols:: $\det$ : determinant, $k$ : integer and $D_{j}$ : quantities
Permalink:: http://dlmf.nist.gov/1.11.E26
Encodings:: TeX, pMML, png

where the column vector $h_{k}^{{(m)}}$ consists of the first $k$ members of the sequence $a_{m},a_{{m-1}},a_{{m-2}},\dots$ with $a_{j}=0$ if $j<0$ or $j>n$ .

Then $f(z)$ , with $a_{n}\not=0$ , is stable iff $a_{0}\not=0$ ; $D_{{2k}}>0$ , $k=1,\dots,\left\lfloor\frac{1}{2}n\right\rfloor$ ; $\mathop{\mathrm{sign}\/}\nolimits D_{{2k+1}}=\mathop{\mathrm{sign}\/}\nolimits a_{0}$ , $k=0,1,\dots,\left\lfloor\frac{1}{2}n-\frac{1}{2}\right\rfloor$ .

§1.11 Zeros of Polynomials

Contents

§1.11(i) Division Algorithm

¶ Horner’s Scheme

¶ Extended Horner Scheme

§1.11(ii) Elementary Properties

¶ Descartes’ Rule of Signs

¶ Example

§1.11(iii) Polynomials of Degrees Two, Three, and Four

¶ Quadratic Equations

¶ Cubic Equations

¶ Example

¶ Quartic Equations

¶ Example

§1.11(iv) Roots of Unity and of Other Constants

§1.11(v) Stable Polynomials

¶ Hurwitz Criterion