unit 4. polynomial

7/31/2019 Unit 4. Polynomial.

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I.E.S. FERNANDO III EL SANTO / PROYECTO BILINGE A.N.L.: MATHS1

MATHS

UNIT 4. POLYNOMIAL

UNIT 4. POLYNOMIAL

A polynomial is an expression of finite length constructed from variables (also known as indeterminates),usually represented by lettres and constants (called the coefficients of the terms), using only theoperations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x

2

4x+ 7 is a polynomial, but x2

4/x+ 7x3/2

is not, because its second term involves division by the variablex (4/x) and because its third term contains an exponent that is not an integer (3/2). The exponent on avariable in a term is called the degree of that variable in that term, the degree of the term is the sum of thedegrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term.For example, x

3- 4x + 2 is a polynomial whose degree is three and the degree of y

3x + 8x

4y

2is six. A term

with no variables is called a constant term, or just a constant. The degree of a (nonzero) constant term is0.

For example:

is a term. The coefficient is 5, the variables are xand y, the degree of xis in the term two, while thedegree of yis one.

The degree of the entire term is the sum of the degrees of each variable in it, so in this example thedegree is 2 + 1 = 3.

Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:

It consists of three terms: the first is degree two, the second is degree one, and the third is degreezero.

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used toform polynomial equations, which encode a wide range of problems, from elementary word problems tocomplicated problems in the sciences; they are used to define polynomial functions, which appear in

settings ranging from basic chemistry and physics to economics and social science; they are used incalculus and numerical analysis to approximate other functions. In advanced mathematics, polynomialsare used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.

Operations with polynomials

The commutative law of addition can be used to freely permute terms into any preferred order. Inpolynomials with one variable, the terms are usually ordered according to degree, either in "descendingpowers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in theexample above is written in descending powers of x. The first term has coefficient 3, variable x, and

exponent 2. In the second term, the coefficient is 5. The third term is a constant. Since the degree of anon-zero polynomial is the largest degree of any one term, this polynomial has degree two.


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Two terms with the same variables raised to the same powers are called "like terms", and they can becombined (after having been made adjacent) using the distributive law into a single term, whosecoefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes

the coefficient 0, in which case their combination just cancels out the terms. Polynomials can be addedusing the associative law of addition (which simply groups all their terms together into a single sum),possibly followed by reordering, and combining of like terms. For example, if

then

which can be simplified to

To work out the product of two polynomials into a sum of terms, the distributivelaw is repeatedly applied, which results in each term of one polynomial beingmultiplied by every term of the other. For example, if

then

which can be simplified to

The sum or product of two polynomials is always a polynomial.

Polynomial long division

Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same orlower degree, a generalized version of the familiar arithmetic technique called long division. It can be doneeasily by hand, because it separates an otherwise complex division problem into smaller ones.

Example: Find


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The problem is written like this:

The quotient and remainder can then be determined as follows:

1. Divide the first term of the numerator by the highest term of the denominator (meaning theone with the highest power of x, which in this case is x). Place the result above the bar (x

3

x= x2).

2. Multiply the denominator by the result just obtained (the first term of the eventual quotient).Write the result under the first two terms of the numerator (x

2 (x 3) = x

3 3x

2).

3. Subtract the product just obtained from the appropriate terms of the original numerator(being careful that subtracting something having a minus sign is equivalent to addingsomething having a plus sign), and write the result underneath ((x

3 12x

2) (x

3 3x

2) =

12x2

+ 3x2

= 9x2

) Then, "bring down" the next term from the numerator.

4. Repeat the previous three steps, except this time use the two terms that have just beenwritten as the numerator.

5. Repeat step 4. This time, there is nothing to "pull down".


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The polynomial above the bar is the quotient, and the number left over (123) is the remainder.

Evaluation of a polynomial consists of assigning a number to each variable and carrying out theindicated multiplications and additions. For example: The evaluation of P(x )= 2x

2 3x +6 when x = -1 is

P(-1) = 2(-1)2

3(-1) + 6 = 2 + 3 + 6 = 11.

The Remainder Theorem

The Remainder Theorem says that the value of the polynomial p(x) at x= ais the same as the remainderyou get when you divide that polynomial p(x) by xa.We can restate the polynomial in terms of the divisor, and then evaluate the polynomial at x= a. But whenx= a, the factor "xa" is just zero! Then evaluating the polynomial at x= agives us:

p(a) = (a a)q(a) + r(a)= (0)q(a) + r(a)= 0 + r(a)= r(a)

But remember that the remainder term r(a) is just a number! So the value of the polynomial p(x) at x= aisthe same as the remainder you get when you divide that polynomial p(x) by xa.

As a concrete example of p, a, q, and r, let's look at the polynomial p(x) = x3

7x 6, and let's divide bythe linear factor x 4 (so a= 4):


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So we get a quotient of q(x) = x2

+ 4x+ 9 on top, with a remainder of r(x) = 30.

Considering the remainder theorem:

p(4) = (4 4)((4)2

+ 4(4) + 9) + 30= (0)(16 + 16 + 9) + 30= 0 + 30= 30

When you are dividing by a linear factor, you don't "have" to use long polynomial division; instead, you canuse synthetic division (Ruffinis rule), which is much quicker. In our example, we would get:

Factoring and Roots of PolynomialsWhat is factoring?

If you write a polynomial as the product of two or more polynomials, you have factored the polynomial.Here is an example:

The polynomials x-3 and are called factors of the polynomial

Note that the degrees of the factors, 1 and 2, respectively, add up to the degree 3 of the polynomial westarted with. Thus factoring breaks up a complicated polynomial into easier, lower degree pieces.

We are not completely done; we can do better: we can factor

We have now factored the polynomial into three linear (=degree 1) polynomials. Linear polynomials arethe easiest polynomials. We can't do any better. Whenever we cannot factor any further, we say we have

factored the polynomial completely.


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Roots of polynomials.An intimately related concept is that of a root, also called a zero, of a polynomial. A number x=ais called a

root of the polynomial f(x), if

Once again consider the polynomial

Let's plug in x=3 into the polynomial.

Consequently x=3 is a root of the polynomial . Note that (x-3) is a factor of

.

Let's plug in into the polynomial:

Thus, is a root of the polynomial . Note that is a

factor of .

Roots and factoring.

This is no coincidence! When an expression (x-a) is a factor of a polynomial f(x), then f(a)=0.Algebraic fraction

An algebraic fraction is the indicated quotient of two algebraic expressions.

Two examples of algebraic fractions are and . Algebraic fractions are subject tothe same laws as arithmetic fractions.


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HISTORY OF ALGEBRA

The history of algebra began in ancient Egypt and Babylon, where people learned tosolve linear (ax= b) and quadratic (ax2 + bx= c) equations, as well as indeterminateequationssuch as x2 + y2 = z2, whereby several unknowns are involved. The ancientBabylonians solved arbitrary quadratic equations by essentially the same procedurestaught today. They also could solve some indeterminate equations.

The Alexandrian mathematicians Hero of Alexandria and Diophantus continued thetraditions of Egypt and Babylon, but Diophantus's book Arithmeticais on a much higherlevel and gives many surprising solutions to difficult indeterminate equations. Thisancient knowledge of solutions of equations in turn found a home early in the Islamic

world, where it was known as the "science of restoration and balancing." (The Arabicword for restoration, al-jabru, is the root of the word algebra.) In the 9th century, the Arabmathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposof the basic theory of equations, with both examples and proofs. By the end of the 9thcentury, the Egyptian mathematician Abu Kamil had stated and proved the basic lawsand identities of algebra and solved such complicated problems as finding x, y, and zsuch that x + y + z = 10, x2 + y2 = z2, and xz = y2.

Ancient civilizations wrote out algebraic expressions using only occasional abbreviations,but by medieval times Islamic mathematicians were able to talk about arbitrarily highpowers of the unknown x, and work out the basic algebra of polynomials (without yetusing modern symbolism). This included the ability to multiply, divide, and find squareroots of polynomials as well as a knowledge of the binomial theorem. The Persianmathematician, astronomer, and poet Omar Khayyam showed how to express roots ofcubic equations by line segments obtained by intersecting conic sections, but he couldnot find a formula for the roots. A Latin translation of Al-Khwarizmi's Algebraappeared inthe 12th century. In the early 13th century, the great Italian mathematician LeonardoFibonacci achieved a close approximation to the solution of the cubic equation x3 + 2x2 +cx= d. Because Fibonacci had traveled in Islamic lands, he probably used an Arabicmethod of successive approximations.

Early in the 16th century, the Italian mathematicians Scipione del Ferro, NiccolTartaglia, and Gerolamo Cardano solved the general cubic equation in terms of theconstants appearing in the equation. Cardano's pupil, Ludovico Ferrari, soon found anexact solution to equations of the fourth degree (see quartic equation), and as a result,mathematicians for the next several centuries tried to find a formula for the roots ofequations of degree five, or higher. Early in the 19th century, however, the Norwegianmathematician Niels Abel and the French mathematician Evariste Galois proved that nosuch formula exists.

An important development in algebra in the 16th century was the introduction of symbolsfor the unknown and for algebraic powers and operations. As a result of this

development, Book III of La gometrie (1637), written by the French philosopher andmathematician Ren Descartes, looks much like a modern algebra text. Descartes's


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most significant contribution to mathematics, however, was his discovery of analyticgeometry, which reduces the solution of geometric problems to the solution of algebraic

ones. His geometry text also contained the essentials of a course on the theory ofequations, including his so-called rule of signs for counting the number of whatDescartes called the "true" (positive) and "false" (negative) roots of an equation. Workcontinued through the 18th century on the theory of equations, but not until 1799 was theproof published, by the German mathematician Carl Friedrich Gauss, showing that everypolynomial equation has at least one root in the complex plane.

By the time of Gauss, algebra had entered its modern phase. Attention shifted fromsolving polynomial equations to studying the structure of abstract mathematical systemswhose axioms were based on the behavior of mathematical objects, such as complexnumbers, that mathematicians encountered when studying polynomial equations. Two

examples of such systems are algebraic groups (see Group) and quaternions, whichshare some of the properties of number systems but also depart from them in importantways. Groups began as systems of permutations and combinations of roots ofpolynomials, but they became one of the chief unifying concepts of 19th-centurymathematics. Important contributions to their study were made by the Frenchmathematicians Galois and Augustin Cauchy, the British mathematician Arthur Cayley,and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions werediscovered by British mathematician and astronomer William Rowan Hamilton, whoextended the arithmetic of complex numbers to quaternions while complex numbers areof the form a + bi, quaternions are of the form a + bi + cj + dk.

Immediately after Hamilton's discovery, the German mathematician HermannGrassmann began investigating vectors. Despite its abstract character, Americanphysicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists,just as Hamilton had recognized the usefulness of quaternions. The widespreadinfluence of this abstract approach led George Boole to write The Laws of Thought(1854), an algebraic treatment of basic logic. Since that time, modern algebraalsocalled abstract algebrahas continued to develop. Important new results have beendiscovered, and the subject has found applications in all branches of mathematics and inmany of the sciences as well.

HISTORY OF ALGEBRA QUESTIONS

1) Where does the word algebra come from?2) Where did the history of algebra begin?3) What kind of equations could ancient Egyptians solve?4) What was a very important development in algebra in the 16th century?5) When did algebra enter its modern phase?

unit 4. polynomial

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