Learning requirements:

A polynomial is arranged in ascending and descending order of letters.

The proposition in this section mainly examines the concepts of algebraic expression, monomial, monomial coefficient and degree, polynomial degree and term number. Polynomials are arranged according to the ascending (descending) power of letters, and most of them appear in the form of filling in the blanks.

Core knowledge

1. The concept of single item

Algebraic expressions 3a, -mn, x2, -abx, 4x3 are all products of numbers and letters. Such algebraic expressions are called monomials, and a single number or letter is also a monomial.

The numerical factor in a single item is called the coefficient of the item.

In a monomial, the sum of the exponents of all the letters is called the degree of the monomial. For example:

3a is the product of 3 and the letter A, and the exponent of the letter A is 1, so the coefficient of the monomial 3a is 3 and the degree is 1.

-mn can be regarded as-1 mn, which is the product of-1 and mn, so the coefficient of monomial -mn is-1 and the degree is 2.

The coefficient of single x2 is 1, and the degree is 2, so the coefficient 1 is usually omitted here.

The coefficient of the monomial -2abx is -2, and the degree is equal to the sum of three letter indices, that is, 1+ 1 = 3. Note that the coefficient of this single item is negative. Pay attention to the coefficient of the single item, including the symbol in front of it, and don't miss it.

According to the definition of monomial, the monomial only includes multiplication (including power) and division of numbers. So algebraic expressions like m2n and-are all monomials. Among them, the monomial-which can be regarded as the product of number-and ab, has a coefficient of-and a degree of 2.

Algebraic expressions with letters in denominator are generally not monomial. For example, they cannot be regarded as the product of numerical factors and letters.

2. The concept of polynomial

The sum of several monomials is called polynomial. For example, algebraic expressions 2a+b, x2-3x+2, m3-3n3-2m+2n are all polynomials. Among them, x2-3x+2 can be regarded as the sum of individual x2, -3x and 2, and m3-3n3-2m+2n can be regarded as m3, -3n3, -2m and 2n.

In polynomials, each monomial is called a polynomial term. Items without letters are called constant items. When determining the polynomial term, we should pay special attention to the symbol of the term, such as

The polynomial x2 -3x+2 * * has three terms, namely x2,-3x and 2. The second term is "-3x" instead of "3x", and 2 is a constant term.

In a polynomial, the degree of the highest term is the degree of the polynomial. For example, 2a+b is a linear binomial; X2-3x+2 is a quadratic trinomial; M3-3n3-2m+2n is a cubic quartile.

Monomial and polynomial are collectively called algebraic expressions. Among them, the monomial only allows multiplication and division with numbers as divisors. Polynomials must contain addition or subtraction, but there can be no letter division.

Therefore, the monomial does not contain addition or subtraction, but the polynomial must contain addition or subtraction, which is the most obvious difference between the two.

3. The arrangement of polynomials

Since a polynomial is the sum of several monomials, additive commutative law and the associative law can be used to exchange the positions of the terms in the polynomial. For the convenience of calculation, polynomials are generally arranged in the order of the exponential size of one of the letters.

Polynomials arranged in descending alphabetical order are called polynomials arranged in descending alphabetical order. Polynomials arranged in descending alphabetical order are called polynomials arranged in ascending alphabetical order.