1 UNIT 4. POLYNOMIAL Operations with Polynimials

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 the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x (4/x) and because its third term contains an exponent that is not an integer (3/2). The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term. For example, x3- 4x + 2 is a polynomial whose degree is three and the degree of y3x + 8x4y2 is six. A term with no variables is called a constant term, or just a constant. The degree of a (nonzero) constant term is 0.

For example:

is a term. The coefficient is –5, the variables are x and y, the degree of x is in the term two, while the degree of y is one.

The degree of the entire term is the sum of the degrees of each variable in it, so in this example the degree 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 degree zero.

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated 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 in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.

Operations with polynimials

The commutative law of addition can be used to freely permute terms into any preferred order. In polynomials with one variable, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example 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 a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.

Two terms with the same variables raised to the same powers are called "like terms", and they can be combined (after having been made adjacent) using the distributive law into a single

1 term, whose coefficient 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 added using 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 distributive law is repeatedly applied, which results in each term of one polynomial being multiplied 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 or lower degree, a generalised version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.

Example: Find

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 the one with the highest power of x, which in this case is x). Place the result above the bar (x3 ÷ 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 (x2 · (x − 3) = x3 − 3x2).

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 adding something having a plus sign), and write the result underneath ((x3 − 12x2) − (x3 − 3x2) = −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 been written as the numerator.

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

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 the indicated multiplications and additions. For example: The evaluation of P(x )= 2x2 – 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 = a is the same as the remainder you get when you divide that polynomial p(x) by x – a. We can restate the polynomial in terms of the divisor, and then evaluate the polynomial at x = a. But when x = a, the factor "x – a" is just zero! Then evaluating the polynomial at x = a gives 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 = a is the same as the remainder you get when you divide that polynomial p(x) by x – a.

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 by the linear factor x – 4 (so a = 4):

4 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 can use synthetic division (Ruffini’s rule), which is much quicker. In our example, we would get:

Factoring and Roots of Polynomials

What 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 we started 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 are the easiest polynomials. We can't do any better. Whenever we cannot factor any further, we say we have factored the polynomial completely.

Roots of polynomials. An intimately related concept is that of a root, also called a zero, of a polynomial. A number x=a is 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 to the 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 to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They also could solve some indeterminate equations.

The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus's book Arithmetica is on a much higher level and gives many surprising solutions to difficult indeterminate equations. This ancient 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 Arabic word for restoration, al-jabru, is the root of the word algebra.) In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. By the end of the 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such 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 high powers of the unknown x, and work out the basic algebra of polynomials (without yet using modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials as well as a knowledge of the binomial theorem. The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations by line segments obtained by intersecting conic sections, but he could not find a formula for the roots. A Latin translation of Al-Khwarizmi's Algebra appeared in the 12th century. In the early 13th century, the great Italian mathematician Leonardo Fibonacci 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 Arabic method of successive approximations.

Early in the 16th century, the Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano solved the general cubic equation in terms of the constants appearing in the equation. Cardano's pupil, Ludovico Ferrari, soon found an exact 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 of equations of degree five, or higher. Early in the 19th century, however, the Norwegian mathematician Niels Abel and the French mathematician Evariste Galois proved that no such formula exists.

An important development in algebra in the 16th century was the introduction of symbols for the unknown and for algebraic powers and operations. As a result of this development, Book III of La géometrie (1637), written by the French philosopher and mathematician René Descartes, looks much like a modern algebra text. Descartes's most significant contribution to mathematics, however, was his discovery of analytic geometry, 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 of equations, including his so-called rule of signs for counting the number of what Descartes called the "true" (positive) and "false" (negative) roots of an equation. Work continued through the 18th century on the theory of equations, but not until 1799 was the proof published, by the German mathematician Carl Friedrich Gauss, showing that every polynomial equation has at least one root in the complex plane.

By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behavior of mathematical objects, such as complex numbers, that mathematicians encountered when studying polynomial equations. Two examples of such systems are algebraic groups (see Group) and quaternions, which share some of the properties of number systems but also depart from them in important ways. Groups began as systems of permutations and combinations of roots of polynomials, but they became one of the chief unifying concepts of 19th-century mathematics. Important contributions to their study were made by the French mathematicians Galois and Augustin Cauchy, the British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions were discovered by British mathematician and astronomer William Rowan Hamilton, who extended the arithmetic of complex numbers to quaternions while complex numbers are of the form a + bi, quaternions are of the form a + bi + cj + dk.

Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann began investigating vectors. Despite its abstract character, American physicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists, just as Hamilton had recognized the usefulness of quaternions. The widespread influence of this abstract approach led George Boole to write The Laws of Thought (1854), an algebraic treatment of basic logic. Since that time, modern algebra—also called abstract algebra—has continued to develop. Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well.

HISTORY OF ALGEBRA QUESTIONS

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