Polynomials

A number represented by algebraic symbols is referred to as an algebraic expression.

An algebraic expression whose parts are not separated by + or – is called a term; as 2×3, –5×yz, and xy/z.

In the expression 2×3 – xyz – xy/z there are three terms. The expression c×(a-b) is a term.

An algebraic expression of one term is known as a monomial or a simple expression. (xy and 3ab are monomials).

An algebraic expression of more than one term is called a polynomial. Such is, for instance, the expression ab – a + b – 10 + (a – b)/c.

In other words we can say that algebraic expressions which consist of several monomials connected by the + and – signs are known as polynomials.

Terms of a polynomial are separate expressions which form the polynomial by the aid of addition and subtraction. Usually, the terms of a polynomial are taken with the signs preceding them; for instance, we say: term –a, term +b², and so on.

A polynomial of two terms is called a binomial, e.g. 3a+2b and x² – y² are binomials. Similarly a+b+c is a trinomial.

Thus, all algebraic expressions are divided into two groups according to the last algebraic operation indicated: monomials and polynomials.

An expression, any term of which is a fraction, is referred to as a fractional expression, as –3x+a/x; all the other expressions are called integral ones.

An algebraic expression such as 5x³–7x² + 9x + 6 is a polynomial or an integral expression in the letter x. It is composed of one or more terms, each of which is either an integral power of x multiplied by a constant or a constant which is free of x. The constant multipliers 5, 7, 9 are called coefficients; the upper numbers: 3, 2 are exponents; 6 is the constant term. The polynomial is of the third degree in x since 3 is the highest exponent appearing in the expression.

An expression such as 2x²y+5x³yz³-9xyz+2x+7 is a polynomial in x, y and z. The degree of a polynomial in several letters is the highest degree that any single term has in those letters. Thus, the above expression is of the seventh degree in x, y and z since the sum of the exponents of the second term is seven.

Let’s consider four fundamental operations of polynomials.

The first operation is addition. In order to add polynomials, you should place them in such a way that like terms fall under each other, and add the coefficients in each column to find the final coefficient of that term.

The second one is subtraction. To subtract one polynomial from another place the terms of the subtrahend under like terms of the minuend, change the signs of the terms of the subtrahend and add.

The third operation is multiplication. Suppose, you have been given two polynomials and have been asked to multiply one of them by the other. In order to do it, you are to multiply each term of one by every term of the other and to add the products thus obtained.

And the last one is division. To divide one polynomial by another, arrange both the dividend and the divisor in ascending or descending powers of some letter common to both and write the quotient as a fraction.

The rule concerning the operation of division may be stated in the following way:

1. Divide the leading term of the dividend by the leading term of the divisor, obtaining the first term of the quotient.

2. Multiply each term of the divisor by this term of the quotient and subtract the product from the dividend.

The remainder found by this subtraction is used as the dividend and the process is repeated. The work is continued until a remainder is reached which is of lower degree than the divisor. In any case of division if the remainder is zero, the division is exact.

Date: 2015-01-11; view: 803