Mathematics
Roots of Polynomials
The roots of a polynomial are the values of the variable that make the polynomial equal to zero. For a polynomial of degree n, there are at most n distinct roots. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots, counting multiplicity.
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10 Key excerpts on "Roots of Polynomials"
eBook - ePubAlgebra & Geometry
An Introduction to University Mathematics
- Mark Verus Lawson(Author)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
x − 12. Find the remaining roots.7. Find all the roots of the following polynomials.(a) x3 + x2 + x + 1.(b) x3 − x2 − 3x + 6.(c) x4 − x3 + 5x2 + x − 6.8. Write each of the following polynomials as a product of real linear or real irreducible quadratic factors.(a) x3 − 1.(b) x4 − 1.(c) x4 + 1.9. *Let λ1 ,…,λr be r real numbers. Construct the Lagrange polynomials pi (x), where 1 ≤ i ≤ r, that have the following property7.5 ARBITRARY ROOTS OF COMPLEX NUMBERSp i(=λ i){1if j = i0if j ≠ i .The number one is called unity.3 A complex number z such that zn = 1 is called an nth root of unity. These are therefore exactly the roots of the equation zn − 1 = 0. We denote the set of nth roots of unity by Cn . We have proved that every non-zero complex number has two square roots. More generally, by the fundamental theorem of algebra, we know that the equation zn − 1 = 0 has exactly n roots as does the equation zn − a = 0 where a is any complex number. The main goal of this section is to prove these results without assuming the fundamental theorem of algebra. To do this, we shall need to think about complex numbers in a geometric, rather than an algebraic, way. We only need Theorem 7.3.3 : every polynomial of degree n has at most n roots.There is also an important definition that we need to give at this point which gets to the heart of the classical theory of polynomial equations. The word radical simply means a square root, or a cube root, or a fourth root and so on. The four basic operations of algebra, that is addition, subtraction, multiplication and division, together with the extraction of nth roots are regarded as purely algebraic operations. Although failing as a precise definition, we say informally that a radical expression is an algebraic expression involving n
eBook - PDFAbstract Algebra
A Gentle Introduction
- Gary L. Mullen, James A. Sellers(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
In [5] the authors provide a reprinted English version of the book Disquisitiones Arithmeticae . Theorem 7.15 (Fundamental Theorem of Algebra) Every polynomial f ( x ) ∈ C [ x ] of degree greater than one has a root in the field C of complex numbers. Using this theorem, we can quickly see that the following result must also hold. Theorem 7.16 The only irreducible polynomials over the complex numbers are of degree one. You should convince yourself why this result holds. 7.3 Exercises 1. Factor the polynomial x 2 + 5 over the field of complex numbers. 2. Factor the polynomial x 4 + 1 over the field of real numbers. 3. Factor the polynomial x 4 − 1 over the field of complex numbers. 7.4 Root formulas In this section, we discuss a few results concerning the problem of solving polynomial equations motivated by the quadratic formula. Given a linear equation � parenleftbigg parenrightbigg Polynomials 119 ax + b = 0 over the real numbers with a = 0, we can find a formula for the solution; b � namely x = a . − Similarly for a quadratic equation ax 2 + bx + c = 0 , the quadratic formula can be employed to show that we have two solutions, − b ± √ b 2 − 4 ac x = . (7.2) 2 a It is important to note that the quadratic formula was, in essence, known to Babylonian mathematicians several thousand years ago. It is instructive to see just how the two roots in the quadratic formula (7.2) arise. The primary tool that is needed is often known as “completing the square.” We begin with the quadratic equation ax 2 + bx + c = 0 where a = 0 . We then factor the constant a from the first two terms of the left-hand side of the equation. b a x 2 + x + c = 0 . a Next, we complete the square by adding a constant term within the paren-theses on the left-hand side of the equation so that the terms within the parentheses form a square. In this instance, the amount to be added is ( 2 b a ) 2 .
eBook - PDFThe Student's Introduction to MATHEMATICA ®
A Handbook for Precalculus, Calculus, and Linear Algebra
- Bruce F. Torrence, Eve A. Torrence(Authors)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
4 Algebra 4.1 Factoring and Expanding Polynomials A polynomial in the variable x is a function of the form: f HxL = a 0 + a 1 x + a 2 x 2 + ∫ + a n x n , where the coefficients a 0 , a 1 , … , a n are real numbers. Polynomials may be expressed in expanded or in factored form. Without a computer algebra system, moving from one form to the other is a tedious and often difficult process. With Mathematica, it is quite easy; the commands needed to transform a polynomial are called Expand and Factor. In[1]:= Clear@f, xD; f @x_D := 12 -3 x -12 x 3 + 3 x 4 In[3]:= Plot@f @xD, 8x, -2, 5
eBook - PDF- P Adams, K Smith;R V??born??;;(Authors)
- 2004(Publication Date)
- WSPC(Publisher)
Chapter 6 Polynomials Polynomial functions have always been important, if for nothing else than because, in the past, they were the only functions which could be readily evaluated. In this chapter we define polynomials as algebraic entities rather than functions, establish the long divi- sion algorithm in an abstract setting, we also look briefly at zeros of polynomials and prove the Taylor Theorem for polynomials in a generality which cannot be obtained by using methods of calculus. 6.1 Polynomial functions If M is a ring and ao, al, a2, . . . , an E M then a function of the form is called a polynomial, or sometimes more explicitly, a polynomial with coefficients in M . Obviously, one can add any number of zero coefficients, or rewrite Equation (6.1) in ascending order of powers of z without changing the polynomial. The domain of definition of the polynomial is naturally M , but the definition of A(x) makes sense for any x in a ring which contains M . This natural extension of the domain of definition is often understood without explicitly saying so. If A and B are two polynomials then the polynomials A + B, -A and AB are defined in the obvious way as A + B : x H A(x) + B(x) -A : x -A(x) AB : z H A(x)B(x) The coefficients of A+B are obvious; they are the sums of the corresponding coefficients of A and B. The zero polynomial function is the zero function, 167 168 Introduction to Mathematics with Maple that is 10 : x H 0. Similarly, the coefficients of -A have opposite signs to the coefficients of A. The coefficients of AB are obtained by multiplying through, collecting terms with the same power of x and sorting them in descending (or ascending) powers of x. If and P = AB, then There is a clear pattern to the formulae + an-ibm (6.2). In order to subsume them in a compact formula we set a k = 0 for k > n and bk = 0 for k > rn. Then we can rewrite Equations (6.2) as k pk = ak-jbj j=O for' k = 1,2,.
eBook - ePub- A. David Wunsch(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
z − 9 we write>> syms z >> factor(z^4-6*z^3+8*z^2+6*z-9)which yields ans = (z − 1)*(z + 1)*(z − 3)^2This clearly shows roots at 1, −1, and 3, the latter being a root of multiplicity 2. The terms in the polynomial do not have to be entered into factor in descending powers of z.Had we sought the roots of the polynomial by using roots, and if we were working in long format, we would have obtained this less satisfactory result:>> roots([1 −6 8 6 −9]) ans = 3.000000086034111 2.999999913965896 −0.999999999999999 0.999999999999999Because of rounding error, we do not clearly see that there are roots at 3, 1, and −1. In fact, roots seems to show that there are four different roots. However, there are a number of reasons not to use factor. This function, factor, will factor polynomials into terms containing (z−zk)nk, where zkis a root and nkits multiplicity only when zkis rational (which of course means it is real as well). The limitations of factor are illustrated in these examples:>> syms z >> factor(z^2-4) ans = (z − 2, z + 2) % the above is done successfully >> factor(z^2-5) ans = z^2 − 5 % the above factorization is not done as it would involve sqrt (5), an irrational. >> factor(z^2+4) ans = z^2 + 4 %t he above factorization is not done because it would involve5.2.4 Derivatives of Polynomials and the Function Rootsi2 which is not real and thus not rational>> factor(z^3-z^2+4*z-4) ans = (z − 1, z^2 + 4) % the above factorization is only partially successful. Thefactors involving ±i2 are not present.It is easy to show that if a polynomial has a zero of multiplicity 1 at zk, then the derivative of this polynomial cannot have a zero there. A polynomial P(z) with a zero of multiplicity one at zkcan be written as P(z) = an(z − zk) Q(z), where Q(z) is a polynomial of degree n − 1 such that Q(zk ) ≠ 0. This follows from Equation 5.3 . The first derivative of this expression is P′(z) = anQ(z) + an(z − zk)Q′(z), which does not vanish at zk
eBook - ePub- T.H Jackson(Author)
- 2023(Publication Date)
- CRC Press(Publisher)
α obtainingg( x )=(hx − α)( x )where h(x) is a polynomial of degree r−1. Apart from α all roots of g have to be roots of h. But since h has degree r−1 it can have at most r−1 roots. So g has at most 1+(r−1)=r.It is of course also possible for a polynomial of degree n>1 to have fewer than n roots in a field containing its coefficients (and possibly none at all). Say (x) ∈ F[x] has degree n and has roots α1 ,α2 ,…,αkin F where k<n. Then in F[x](2.14)f( x )=(…x −)α 1(gx −)α k( x )where g(x) ∈ F[x] has no linear factors and so is of degree 2 or more. As in the first paragraph of this section, we can construct an extension field K of F containing a root of g. Since it contains F, the new field K will contain all the previous roots α1 ,α2 ,…,αkas well as at least one root, say αk+1 , of g (which will also be a root of f). If g does not have as many roots in K as its degree allows, then in K[x] the polynomial g(x) will factorize as(2.15)g( x )=(… hx −)αk + 1( x )where h(x) ∈ K[x] has no linear factors. This impliesf( x )=(…x −)α 1(x −)α k(x −α
No longer available |Learn moreFoundations of Mathematics
Algebra, Geometry, Trigonometry and Calculus
- Philip Brown(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
discriminant. More about this later.EXAMPLE 3.3.5. We can solve the equation −4x2 − 13x + 7 = 0 by substituting a = −4, b = −13, and c = 7 in the quadratic formula, that is,3.4POLYNOMIALS
Polynomials and the algebraic operations that can be applied to them are the main concern of this chapter. We will begin with the formal definition of a polynomial (in one variable) and then explain how polynomials can be added, subtracted, multiplied, and divided.DEFINITION 3.4.1. A polynomial in x is a sum of the formanxn+ an−1xn−1+…+a1 x+a0 ,where n is a nonnegative integer and each coefficient aj for j = 1, … n is a real number. If an is not equal to zero, then the polynomial is said to be of degree n and an is called the leading coefficient of the polynomial.EXAMPLE 3.4.1. In table 3.1 are a few polynomials and their corresponding degrees.TABLE 3.1. PolynomialsDEFINITION 3.4.2. Polynomials of degree 2, 3, 4, and 5 are called quadratic, cubic, quartic, and quintic polynomials, respectively.DEFINITION 3.4.3. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial.EXAMPLE 3.4.2. x − 3x3 is a binomial, and is a trinomial.3.4.1Addition and Subtraction of Polynomials
Polynomials are added or subtracted by adding like powers of x. Remember your school teacher saying “add apples to apples, bananas to bananas”, and so on!EXAMPLE 3.4.3.(i) (ii) 3.4.2Multiplication of Polynomials
We have already seen an example of polynomial multiplication using the foil rule in section 1.8.1
eBook - PDFTopics In Polynomials: Extremal Problems, Inequalities, Zeros
Extremal Problems, Inequalities, Zeros
- Gradimir V Milovanovic, Themistocles M Rassias, D S Mitrinovic(Authors)
- 1994(Publication Date)
- World Scientific(Publisher)
C H A P T E R 1 General Concepts of Algebraic Polynomials 1.1. P O L Y N O M I A L S A N D E Q U A T I O N S 1.1.1. Preliminaries Let F be a field. The ring of polynomials in x over F, denoted by F[i], is defined to be the set of all formal expressions P(x) = aa + atx + + a n x n , where the a;, the coefficients of the polynomial P in x, are elements in F. If a n ^ 0, then the degree of P, denoted by dg P, is n. Therefore, the degree of a polynomial P is the highest power of x that occurs in the expression for P(x) with a non-zero coefficient. In F[i] we define equality, sum, and product of two polynomials as follows. Given two polynomials P(x) = a 0 + a 1 x + --- +a n x n and Q(x) = b 0 + b i x + ---+b m x m , then the polynomials are equal if, and only if, a; = 6; for all i > 0, that is when their corresponding coefficients are equal. We define the sum of P and Q by {P + Q)(x) = P(x) + Q(x) = c 0 + ax + • • • + c r x r , l 2 GENERAL CONCEPTS OF POLYNOMIALS where for each t, C; = a; + (>;, and the product of P and Q by (PQ)(x) - P(x)<30r) = co + cia; + • + c,*', where c,- = a,6 0 + <*i-i&i + h ai&i_i + a 0 b„ for every i . Then, one can show that F[x] is a commutative ring with unit. It is easy to prove that if P(x) and Q(x) are non-zero elements of V[x], then dg(PQ) = dgP + dgt?. Also, if P{x),Q{x) e F[x] and P(x) + Q(x) ± 0, then dgtP + g j ^ m a x I d g ^ d g Q } . We do not assign a degree to 0{x) = Ox + 0 x n _ 1 + • • • + Ox + 0. The polynomials of degree 0 are called the constants (these are the elements of the field F). Let k ,... , k m be non-negative integers and fc = ( * i , . . . , A m ) , = + + x k = »f» • * * » .
- Alan Sultan, Alice F. Artzt(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
has no rational roots .(c) It is clear thatp ( 2 ) = 0 .(d) Since there are no rational roots, 2 , which is a root of p (x ), cannot be rational.Can you see how powerful this technique is in proving that a number is irrational? Let us try another example and show that7 3is irrational. Since7 3satisfies the polynomial p (x ) = x 3 − 7 = 0, and the only rational roots possible for this equation are ±7 and ±1, none of which work,7 3is irrational. Similarly, we can show15 4is irrational, or evenA nwhere A is an integer that is not a perfect n th power. Even more elaborate numbers likecan be shown to be irrational in a similar manner. For example, if we let1 + 23x =and cube both sides we get1 + 23, and subtracting 1 from both sides and squaring, we get that (x 3 − 1)2 = 2 or that x 6 − 2x 3 − 1 = 0. The only rational roots of this are ±1 by the rational root theorem, and none of them works. So this equation has no rational roots. Butx 3= 1 + 2is a root of this equation, so it must be irrational. What a nice tool the rational root theorem is!1 + 23We have just seen that several irrational numbers can be obtained as Roots of Polynomials. For example, 2 is a root of the polynomial p (x ) = x 2 − 2, andis a root of p (x ) = x 6 − 2x 3 − 1. It is a natural question to ask if all irrational numbers are Roots of Polynomials with integer coefficients. For a while, many people believed that. But to allow for the possibility that this was not true, mathematicians defined the term algebraic number.1 + 23An algebraic number is a number that is the root of a polynomial with integral coefficients. Thus 2 andand1 + 2315 4are algebraic as we saw in the last two paragraphs. A number that is not algebraic is called transcendental . Thus, a transcendental number is a number that is not a root of any polynomial with integral
eBook - ePub- Jean-Pierre Tignol(Author)
- 2015(Publication Date)
- WSPC(Publisher)
r = 1 being trivial. By the induction hypothesis,P = P1 …Pr −1 Qfor some polynomial Q. Since Pr divides P, Lemma 5.9 shows that Pr divides Q, so that P1 …Pr divides P.5.4Roots
As in the preceding sections, F denotes a field. For any polynomialP = a0 + a1 X + ··· + an Xn ∈ F[X]we denote by P(·) the associated polynomial functionP(·): F → Fwhich maps any x ∈ F to P(x) = a0 +a1 x+…+an xn . It is readily verified that for any two polynomials P, Q ∈ F[X] and any x ∈ F,(P + Q)(x) = P(x) + Q(x)and(P · Q)(x) = P(x) · Q(x).Therefore, the map P P(·) is a homomorphism from the ring F[X] to the ring of functions from F to F.Definition 5.11. An element a ∈ F is a root of a polynomial P ∈ F[X] if P(a) = 0.Theorem 5.12. An element a ∈ F is a root of a polynomial P ∈ F[X] if and only if X − a divides P.Proof. Since deg(X − a) = 1, the remainder R of the division of P by X − a is a constant polynomial. Evaluating at a the polynomial functions associated with each side of the equationP = (X − a)Q + Rwe getP(a) = (a − a)Q(a) + R,hence P(a) = R. This shows that P(a) = 0 if and only if the remainder of the division of P by X − a is 0. The theorem follows, since the last condition means that X − a divides P.Corollary 5.13. Let P ∈ F[X] be a polynomial of degree 2 or 3. Then P is irreducible over F if and only if it has no roots in F.Proof. This readily follows from Theorem 5.12, since the hypothesis on deg P implies that if P is not irreducible, then it has a factor of degree 1, hence of the form X − a.Definitions 5.14. The multiplicity of a root a of a nonzero polynomial P is the exponent of the highest power of X − a which divides P. Thus, the multiplicity is m if (X − a)mdivides P but (X − a)m+1 does not divide P. A root is called simple when its multiplicity is 1; otherwise it is called multiple
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