Concept is the only criterion. If you don't know the concept, or have a little knowledge, or seem to know it, then you will have no clue, or even have no way to start. Next, we will deeply explain the relationship among monomial, polynomial and algebra, and restore their true features, so that you can see clearly and clearly, and it is no longer a fog. ...
What is a monomial? The product of numbers and letters, such an algebraic expression is a monomial. Among them, a single number or a single letter is also monomial.
Although this definition seems clear, it is also confusing without careful analysis. So, let's extract the key points in this definition and analyze them carefully:
1, the objects in the monomial are numbers and letters:
It is not difficult to see from the definition that there are only numbers in the monomial, such as 36, 1.2, 1/6 (one sixth), Wu (π), 36 Wu, etc. These are monomials; Or just letters, such as a, b, c, ABC, etc. , is a monomial; Either there are numbers and letters, such as 36a, 1.2abc, etc. These are monomials. In other words, the participants in the monomial are numbers, or letters, or numbers and letters.
Numbers include integers, fractions and decimals. It is worth emphasizing that because decimals are included. Of course, it also includes irrational numbers. Because decimals include infinite acyclic decimals, which are irrational numbers. For example, the pi "Wu" we say is an irrational number, because the participants in the monomial also include irrational numbers, so things like "3 Wu B" are also monomials.
2. There is only one operation symbol in the monomial, and that is the multiplication symbol:
We have all studied the concept of algebra, and combined with the concept of monomial, it is not difficult to see that monomial is also a kind of algebra. We also know that the concept of algebraic expression emphasizes the operation symbol, that is, as long as it is an operation symbol and meets other conditions, it is algebraic expression.
The monomial emphasizes one of the operation symbols, namely the multiplication operation symbol. In other words, there is only multiplication in the single item, and only product in the single item!
Judge whether the following algebraic expression is a monomial or not, and why?
Obviously, these four questions are not, because there are not only product relations but also other relations in these four algebraic expressions, so they are not.
Key point: you must not use the letter of the monomial as the denominator, that is, as long as there are letters on the denominator, it is definitely not a monomial.
3. Special items:
If such a problem arises, it is estimated that many friends are ignorant. In fact, it is very simple. Just remember that "the absolute value of a single item is still a single item!" It's that simple.
So, is the following question monomial? Why?
The correct answer will be announced in the next class ~
4. Single coefficient:
In the monomial, there is a part called coefficient.
Because the single item is a product relationship, all parties are called factors. For example, in monomial 3a, 3 and a are both factors. Among them, the numerical factor is called the single coefficient. Therefore, 3 is the coefficient of the monomial 3a. Another example: -8ab, in this monomial, -8 is the coefficient.
Key points about coefficient:
A In a monomial with numbers, the number part is the coefficient of the monomial, and the original symbol of the number part is also a part of the coefficient.
B In a single item without numbers, the coefficient in the single item is either 1,-1, 1 or-1, which mainly depends on the positive or negative of the single item. For example, the coefficient of the monomial "ab" is1; Another example is that the coefficient of the monomial "-bc" is "-1"
C, in a monomial with only numbers, its coefficient is the number and its own sign.
If the above topics are understood, then I will give you a few more questions to see if you understand the coefficients of the monomial:
The answers to these questions will be explained in the next class ~
5, the number of monomials:
In the monomial, there is a part called degree.
In fact, the number of times in the monomial is easy to understand, which is the exponential sum of all letters in the monomial.
So, how do you understand this sentence? In other words, the number of monomials is only related to letters, not to numerical factors. The degree of a monomial is not the exponent of a letter, but the sum of the exponents of all the letters in the monomial. No matter where the letters are in the monomial, as long as they are letters, the sum of their indices is the number of monomials.
Emphasize two points:
A. The number of monomials is only related to letters, not numbers. For example, in the first question above, although there is an index 3 above 6, the index above the number does not participate in the calculation of the number of monomials.
B the number of times of a single non-zero number is 0. Just remember this one. Like the second question above, the number of 8 is "0"
Seeing this, I believe everyone already knows the monomial like the back of his hand.
6. Classification of individual items:
According to the concept of monomials, monomials can be divided into five categories.
A, monomial, is a monomial; For example, 88, pi and 3.33 ... can be rational numbers or irrational numbers;
B, the product between numbers, is a monomial; For example, 3x8x6 is a monomial.
C, a single letter, is a monomial; Such as a, b, c ......
D, the product between letters is a monomial; Such as abc and bd ......
E, the product of numbers and letters is a monomial; Such as 3a, 2cf ......
What is a polynomial? In fact, if you understand the monomial, then the polynomial is better understood.
The sum of several monomials is called polynomial. Is it easy to understand?
It is not difficult to see from the concept of polynomial that polynomial is composed of monomials, and the relationship between monomials in polynomial is "sum".
Concept is the only criterion for judgment. Then, using the concepts of monomial and polynomial, we distinguish 3a and 3+a in the graph:
3a represents the product of a number and a letter, which accords with the concept of monomial. Obviously, 3a is a monomial. In 3+b, the single number 3 is a monomial, and the single letter B is a monomial. The plus sign "+"indicates that the relationship between these two monomials is a "sum" relationship, so it satisfies the concept of polynomial, so 3+b is a polynomial.
In polynomials, there are several definitions that need to be clarified:
1, polynomial term:
In polynomials, each monomial is called a polynomial term. For example, in the polynomial 3+a, both 3 and a are called polynomial terms.
2, the number of polynomials:
The degree of polynomial is the degree of the single item with the highest degree in polynomial. In other words, the degree of polynomial is determined by the monomial with the highest degree in polynomial. As long as we know the degree of monomial, the degree of polynomial is easy to understand. For example, the degree of polynomial 3ab+6d is "2", because in this polynomial, the degree of monomial 3ab is the highest, which is also the degree of polynomial.
3. Number of terms of polynomial:
This is easier to understand. That is, a polynomial has several monomials, that is, the number of terms of a polynomial. For example, this polynomial 6+ab+c has three monomials, so its term number is 3.
So much for polynomials. In order to better understand the concept of polynomial, we have several questions, as shown in the figure below. You can do it if you are interested. We will announce the answer next class ~
What is algebraic expression? If we understand monomials and polynomials, algebraic expressions will be better understood. Because algebraic expression is a general term for monomial and polynomial.
The relationship between 1, monomial, polynomial and algebraic expression;
A monomial is a product of numbers and letters, a polynomial is a monomial, and an algebraic expression is a generic term for both monomials and polynomials.
2, the relationship between algebra and algebra:
Algebra includes algebra, and algebra is a part of algebra. Because algebraic expressions are generic terms for monomials and polynomials, monomials and polynomials are also algebraic.
3. The relationship between algebraic expression and rational expression:
Rational expression includes algebraic expression, which is a part of rational expression. Rational expression is a general term for algebraic expression and fractional expression. Because algebraic expressions are generic terms for monomials and polynomials, monomials and polynomials are also rational numbers.
4, the difference between algebra and fraction:
As shown in the above figure, the denominator of algebraic expression cannot have letters, but the denominator of fraction has letters. The essential difference between algebra and fraction is the position of letters. Under rational conditions, fractions are letters that appear on the denominator, and algebraic expressions are letters that do not appear on the denominator.
5, the difference between rational and irrational:
The difference between rational expression and irrational expression lies in the position where the letters appear. Under algebraic conditions, if a letter appears in the root sign, it is an irrational number, and the other is a rational number.
6. The relationship between rationality, irrationality and algebra:
Algebraic expression is a general term for rational expression and irrational expression. The relationship between the various as shown in the figure below: