POLYNOMIALS. OPERATIONS 1

The order of monomials

The monomials that make up a polynomial can appear in any order. However, they are most commonly ordered by degree, from largest to smallest.

Subtracting polynomials

Subtracting is a special type of addition; therefore, the result of subtracting two polynomials is another polynomial.

P – Q = P + (–Q )

The polynomial –Q is obtained by changing the sign for all the Q monomials.

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Comparing polynomials to integers.

Problem solved

1 Given the polynomials:

Basic terminology

As you already know, the following expression is a polynomial:

It is made up of four monomials: 2x 5; 3/7x 3; – 3 x 2 and 2.7, the degrees of which are 5, 3, 2 and 0, respectively (2.7 has a degree of 0, since 2.7 = 2.7 · x0).

The degree of the polynomial is 5 (that of the monomial with the highest degree).

The variable, x, is also called the indeterminate.

The numbers 2, 3/7, – 3 and 2.7 are the coefficients. As there is no thirddegree monomial, we can say that its coefficient is 0.

In a polynomial, the coefficients are any real numbers.

Operations with polynomials. Sum and product

The result of adding or multiplying two polynomials is another polynomial. These operations have certain properties that allow us to perform operations easily and smoothly*. Let’s take a look: product could have been found by placing one polynomial underneath the other and placing each term of each product underneath the similar term:

Practise adding, subtracting and multiplying polynomials.

Division of polynomials

The result of adding, subtracting or multiplying two polynomials is another polynomial. However, this is not true for division: in general, the quotient of two polynomials is not a polynomial.

Let’s take a look:

Remember how to divide two polynomials, and how we interpret the result. Let’s divide

Remember

Degree of the dividend, m

Degree of the divisor, n m ≥ n

Degree of the quotient: m – n

Degree of the remainder < n Integer division in Z 89 = 5 · 17 + 4, or 5 89 17 5 4 =+

The result of the division is not an integer.

The result can be written in one of the following two ways:

This way of dividing polynomials, where as well as the quotient there is a remainder other than zero, is called integer division.

When the remainder is zero, we say that it is an exact division.

For example, let’s divide the anayaeducacion.es

Practise dividing polynomials.

Breaking down into factors

P (x) = x 4 + x 3 – 10x 2 – 4x + 24 has been broken down into a product of two factors: P (x) = (x 2 – 4)(x 2 + x – 6). And each of these factors can also be broken down: x 2 – 4 = (x – 2)(x + 2) x 2 + x – 6 = (x – 2)(x + 3)

Therefore:

P (x) = (x – 2)(x + 2)(x – 2)(x + 3) = = (x – 2)2 (x + 2)(x + 3)

We need to try and break down all polynomials using this method.

As it is an exact division, the dividend can be given as the product of two factors:

Polynomials, like integers, can be broken down into products of factors. When we try doing this, we face the following problem:

Given a polynomial P (x), how can we work out which other polynomial to divide it by for the division to be exact?

When working with integers, we used the divisibility criteria to answer this question. However, for polynomials,we will use something distantly related.

To find the divisor we need, we will have to perform many divisions, mostly using expressions of the type x – a. The next section will teach us to do this.