A rational expression in algebraic expression. If there is no division or fraction, if there is a division and fraction, but there is no variable in the division or denominator, it is called an algebraic expression.
Algebraic expressions can be divided into definitions and operations, definitions can be divided into monomials and polynomials, and operations can be divided into addition, subtraction, multiplication and division.
Addition and subtraction involve merging similar items. Multiplication and division include basic operations, rules and formulas. Basic operations can be divided into power operations. Rules can be divided into algebra and division, and formulas can be divided into multiplication formula, zero exponential power and negative integer exponential power.
Algebraic expressions and similar items
1. Single item
(1) The concept of monomial: The algebraic expression of the product of numbers and letters is called monomial, and a single number or letter is also a monomial.
Note: Numbers and letters have a product relationship.
(2) Single factor: The letter factor in a single item is called the single factor.
If a monomial contains only letter factors, the coefficient of the positive monomial is 1 and the coefficient of the negative monomial is-1.
(3) The number of monomials: The sum of the indices of all the letters in the monomials is called the number of monomials.
2.polynomial
The concept of (1) polynomial: The sum of several monomials is called polynomial. In polynomials, each monomial is called a polynomial term, and the term without letters is called a constant term. Polynomials with several terms are called polynomials. The symbols in polynomials are regarded as the natural symbols of each term.
(2) Degree of Polynomial: The degree of the term with the highest degree in the polynomial is the degree of the polynomial.
(3) the arrangement of polynomials:
1. Sorting polynomials in descending alphabetical order is called sorting polynomials in descending alphabetical order.
2. Arranging a polynomial according to the exponent of a letter from small to large is called arranging polynomials according to the ascending power of this letter.
Since a polynomial is the sum of several monomials, the position of each term can be exchanged by the addition algorithm, while keeping the value of the original polynomial unchanged.
In order to facilitate the calculation of polynomials, a polynomial is usually arranged in a neat and simple form in a certain order, which is the arrangement of polynomials.
Pay attention to when doing polynomial arrangement problems:
(1) Since a single item contains its preceding attribute symbol, the attribute symbol of each item should still be regarded as a part of the item and moved together.
(2) The arrangement of polynomials with two or more letters should pay attention to:
A. first of all, it must be arranged according to the index of which letter.
B. decide whether to arrange according to this letter.
(3) Algebraic expression:
Monomial and polynomial are collectively called algebraic expressions.
(4) the concept of similar items:
Items with the same letters and times are called similar items, and several constant items are also called similar items.
When mastering the concept of similar items, we should pay attention to:
1. To judge whether several monomials or terms are similar, two conditions must be mastered:
(1) contains the same letters.
The same letter has the same number of times.
2. Similar items have nothing to do with coefficient or alphabetical order.
3. Several constant terms are similar.
(5) Merge similar items:
1. The concept of merging similar projects:
Merging similar terms in polynomials into one term is called merging similar terms.
2. Rules for merging similar projects:
The coefficients of similar items are added together, and the results are taken as coefficients, and the indexes of letters and letters remain unchanged.
3. To merge similar projects:
(1). Find similar projects accurately.
(2) Reverse the distribution law, add the coefficients of similar items together (enclosed in brackets), and keep the letters and their indices unchanged.
(3) Write the merged result.
When mastering the merger of similar projects, we should pay attention to:
1. If the coefficients of two similar items are opposite, the result after merging similar items is 0.
2. Don't leave out items that can't be merged.
3. As long as there are no more similar items, it is the result (either a single item or a polynomial).
The key to merging similar items: correctly judging similar items.
Algebraic expression and multiplication of algebraic expression
Algebraic expressions can be divided into definitions and operations, definitions can be divided into monomials and polynomials, and operations can be divided into addition, subtraction, multiplication and division.
Addition and subtraction involve merging similar items. Multiplication and division include basic operations, rules and formulas. Basic operations can be divided into power operations. Rules can be divided into algebra and division, and formulas can be divided into multiplication formula, zero exponential power and negative integer exponential power.
The power rule of the same base: multiply with the power of the same base and add with the index of the same base.
Power law: power, constant basis, exponential multiplication.
Power law of product: the power of product is equal to the power obtained by multiplying the factors of product respectively and then multiplying them.
There are the following rules for multiplying monomials with monomials: multiply monomials with their coefficients and the same base respectively, and the remaining letters, together with their exponents, remain unchanged as the factorial of the product.
There are the following rules for the multiplication of monomial and polynomial: the multiplication of monomial and polynomial is to multiply each term of polynomial with monomial, and then add the products.
Polynomial and polynomial multiplication have the following rules: polynomial and polynomial multiplication, first multiply each term of one polynomial with each term of another polynomial, and then add the obtained products.
Square difference formula: the product of the sum of two numbers and the difference between these two numbers is equal to the square difference between these two numbers.
Complete square formula: the square of the sum of two numbers is equal to the sum of the squares of these two numbers, plus twice the product of these two numbers. The square of the difference between two numbers is equal to the sum of the squares of these two numbers, MINUS twice the product of these two numbers.
Same base powers divides, the base remains the same, and the exponent is subtracted.
On the learning points of algebraic expressions
Tu Xinmin
Algebraic formula is the most basic formula in algebra, so it is necessary to introduce algebraic formula and learn the following contents (such as fractions, quadratic equations with one variable, etc.). ). On the basis of studying rational number operations, simple algebraic expressions, linear equations and inequalities, algebraic expressions are introduced. In fact, the relevant contents of algebraic expressions have been learned in the sixth grade, but now the contents of algebraic expressions are more applicable than in the past, which increases the background of practical application.
Block diagram of knowledge structure in this chapter:
There are many knowledge points in this chapter that are important or difficult. The key points and difficulties are as follows.
First, the four operations of algebraic expressions
Addition and subtraction of 1. algebraic expressions
Merging similar items is the key and difficult point. When merging similar items, we should pay attention to the following three points: ① Only by mastering the concept of similar items can we distinguish similar items and accurately grasp the two standard letters and letter indexes for judging similar items; (2) The definition of merging similar terms means merging similar terms in polynomials into one term. After merging similar terms, the number of terms in the formula will be reduced, thus simplifying the polynomial; (3) "Merging" refers to adding the coefficients of similar items, and the obtained results are used as new coefficients, and the letters and letter indexes of similar items should remain unchanged.
2. Multiplication and division of algebraic expressions
The emphasis is on multiplication and division of algebraic expressions, especially multiplication formulas. It is difficult for students to master the structural characteristics of multiplication formula and the broad meaning of letters in the formula. Therefore, the flexible application of multiplication formula is difficult, and the handling of symbols in brackets is another difficulty when adding (or removing) brackets. Parentheses (or brackets) are the deformation of polynomials, which should be carried out according to the law of parenthesis (or brackets). In the multiplication and division of algebraic expressions, the single multiplication and division is the key, because the multiplication and division of general polynomials should be "transformed" into the single multiplication and division.
The main problems of the four operations of algebraic expressions are:
Four operations of (1) monomial
This kind of questions mostly appear in the form of multiple-choice questions and application questions, which are characterized by examining four operations of monomials.
(2) Operation of monomial and polynomial
This kind of problems mostly appear in the form of solving problems, which are highly skilled and characterized by examining the four operations of monomials and polynomials.
Second, factorization.
The difficulty is the four basic methods of factorization (raising common factor, using formula, grouping factorization and cross multiplication). Factorization is the reverse deformation of algebraic expression multiplication, and the introduction of factorization should firmly grasp this point.
Polynomial polynomial
A formula consisting of the sum of several monomials is called a polynomial (in subtraction, subtracting a number equals adding its inverse). Each monomial in a polynomial is called a polynomial term, and the highest degree of these monomials is the degree of this polynomial. Items without letters are called constant items. If the degree of the highest term in a formula is 5 and the formula consists of three monomials, it is called a quintic trinomial.
In a broader definition, the sum of 1 or 0 monomials is also a polynomial. According to this definition, polynomials are algebraic expressions. In fact, no theorem is valid only for narrow polynomials, but not for monomials: when 0 is a polynomial, the degree is negative infinity.
Edit this polynomial history
The study of polynomials originated from "solving algebraic equations" and is one of the oldest mathematical problems. Some algebraic equations, such as x+ 1=0, are considered to have no solution before accepting negative numbers. Other polynomials, such as f(x)=x? +1 No root-strictly speaking, there is no real root. If we allow complex numbers, then both real polynomials and complex polynomials have roots, which is the basic theorem of algebra.
Whether the roots of polynomials can be expressed by radical solutions has been the main subject of European mathematics after the Renaissance. The roots of univariate quadratic polynomials are relatively easy. The roots of cubic polynomials need to be represented by complex numbers, even the real roots of real polynomials. The same is true of quartic polynomials. After many years, mathematicians still can't find a general method to solve quintic polynomials with roots. Finally, in 1824, Abel proved that this general method does not exist, which shocked many people. Some years later, Galois introduced the concept of group, and proved that there is no general method to solve polynomials of degree five or more with roots, and its theory was extended to Galois theory. Galois's theory also proves that it is impossible to bisect the ancient Greek puzzle angle. Another problem, the impossible proof of turning a circle into a square, is also related to polynomials. The center of the proof is that pi is a transcendental number, that is, it is not the root of a rational polynomial.
Edit the polynomial function and the root of the polynomial in this paragraph.
Polynomial f∈R[x 1, ..., xn] and an R- algebra a. For (a 1...an)∈An, we use aj instead of xj in F to get an element in A, which is denoted as f(a 1 ... an). In this way, f cAn be regarded as a function from an to a.
If f(a 1...an)=0, then (a 1...an) is called the root or zero of f.
Such as f=x2+ 1. If x is a real number, a complex number, or a matrix, then f has no roots, two roots, and infinite roots!
For example, f=x-y If x is a real number or a complex number, then the zero set of f is the set of all (x, x), which is an algebraic curve. In fact, all algebraic curves come from this.
Edit the basic theorem of algebra in this paragraph.
The basic theorem of algebra means that all unary n-degree (complex) polynomials have n (complex) roots.
Edit the geometric properties of this polynomial.
Polynomial is a simple continuous function, it is smooth, and its differential must be polynomial.
The spirit of Taylor polynomials is to approximate a smooth function with polynomials, and all continuous functions in a closed interval can be written as uniform limits of polynomials.
Edit the polynomial on any ring in this paragraph.
Polynomials can be extended to the case where the coefficients are in any ring. Please refer to the polynomial ring project.
operation sequence
Multiply first and then divide,
After addition and subtraction.
Without brackets,
Do it first.
Peer operation,
From left to right.
Master the operation sequence
Not busy!
Definition: Convert a polynomial into the product of several algebraic expressions. This deformation is called factorization of this polynomial, and it is also factorization.
Significance: It is one of the most important identical deformations in middle school mathematics. It is widely used in elementary mathematics and is a powerful tool for us to solve many mathematical problems. Factorization is flexible and ingenious. Learning these methods and skills is not only necessary to master the content of factorization, but also plays a very unique role in cultivating students' problem-solving skills and developing their thinking ability. Learning it can not only review the four operations of algebraic expressions, but also lay a good foundation for learning scores; Learning it well can not only cultivate students' observation ability, attention and calculation ability, but also improve students' comprehensive analysis and problem-solving ability.
Factorization and algebraic expression multiplication are inverse deformations.
Edit this factorization method.
There is no universal method for factorization. In junior high school mathematics textbooks, the methods of improving common factor, using formulas, grouping factorization and cross multiplication are mainly introduced. There are split addition method, undetermined coefficient method, binary multiplication method, rotational symmetry method, residue theorem method and so on.
Basic method of editing this paragraph
(1) common factor method
The common factor of each term is called the common factor of each term of this polynomial.
If every term of a polynomial has a common factor, we can put forward this common factor, so that the polynomial can be transformed into the product of two factors. This method of decomposing factors is called the improved common factor method.
Specific methods: when all the coefficients are integers, the coefficients of the common factor formula should take the greatest common divisor of all the coefficients; The letter takes the same letter of each item, and the index of each letter takes the smallest number; Take the same polynomial with the lowest degree.
If the first term of a polynomial is negative, a "-"sign is usually put forward to make the coefficient of the first term in brackets become positive. When the "-"sign is put forward, the terms of the polynomial should be changed.
For example:-am+BM+cm =-m (a-b-c);
a(x-y)+b(y-x)= a(x-y)-b(x-y)=(x-y)(a-b)。
⑵ Use the formula method.
If the multiplication formula is reversed, some polynomials can be decomposed into factors This method is called formula method.
Square difference formula: A2-B2 = (a+b) (a-b);
Complete square formula: a22ab+b 2 = (ab) 2;
Note: Polynomials that can be decomposed by the complete square formula must be trinomial, two of which can be written as the sum of squares of two numbers (or formulas), and the other is twice the product of these two numbers (or formulas).
Cubic sum formula: A3+B3 = (a+b) (A2-AB+B2);
Cubic difference formula: A3-B3 = (a-b) (A2+AB+B2);
Complete cubic formula: a 3 3a 2b+3ab 2 b 3 = (a b) 3.
For other formulas, please refer to the above figure.
For example: A 2+4A B+4B 2 = (A+2B) 2 (see the right figure).
Methods to be mastered in editing this paragraph in junior middle school
⑶ Grouping decomposition method
(4) Methods of splitting and supplementing projects
This method refers to disassembling one term of a polynomial or filling two (or more) terms that are opposite to each other, so that the original formula is suitable for decomposition by improving the common factor method, using the formula method or grouping decomposition method. It should be noted that the deformation must be carried out under the principle of equality with the original polynomial.
For example: bc(b+c)+ca(c-a)-ab(a+b)
=bc(c-a+a+b)+ca(c-a)-ab(a+b)
= BC(c-a)+ca(c-a)+BC(a+b)-ab(a+b)
=c(c-a)(b+a)+b(a+b)(c-a)
=(c+b)(c-a)(a+b)。
You can also see the picture on the right.
5] Matching method
For some polynomials that cannot be formulated, they can be fitted in a completely flat way, and then factorized by the square difference formula. This method is called matching method. It belongs to the special case of the method of splitting items and supplementing items. It should also be noted that the deformation must be carried out under the principle of equality with the original polynomial.
For example: x 2+3x-40
=x^2+3x+2.25-42.25
=(x+ 1.5)^2-(6.5)^2
=(x+8)(x-5)。
You can also see the picture on the right.
[6] Cross multiplication.
There are two situations in this method.
① factorization of x2+(p+q) x+pq formula.
The characteristics of this kind of quadratic trinomial formula are: the coefficient of quadratic term is1; Constant term is the product of two numbers; The coefficient of a linear term is the sum of two factors of a constant term. So we can directly decompose some quadratic trinomial factors with the coefficient of1:x 2+(p+q) x+pq = (x+p) (x+q).
② Factorization of KX2+MX+N formula
If k=ac, n=bd, ad+bc=m, then kx 2+MX+n = (ax+b) (CX+d).
The chart is as follows:
A b
×
c d
For example, because
1 -3
×
7 2
And 2-2 1=- 19,
So 7x 2- 19x-6 = (7x+2) (x-3).
Formula of cross multiplication: head-tail decomposition, cross multiplication and summation.
General steps of polynomial factorization:
(1) If the polynomial term has a common factor, then the common factor should be raised first;
(2) If there is no common factor, try to decompose it by formula and cross multiplication;
(3) If the above methods cannot be decomposed, you can try to decompose by grouping, splitting and adding items;
(4) Factorization must be carried out until every polynomial factorization can no longer be decomposed.
It can also be summarized in one sentence: "First, look at whether there is a common factor, and then look at whether there is a formula. Try cross multiplication, and group decomposition should be appropriate. "
Several examples
1. Decomposition factor (1+y) 2-2x2 (1+y 2)+x 4 (1-y) 2.
Solution: The original formula = (1+y) 2+2 (1+y) x2 (1-y)+x4 (1-y) 2-2 (1+y).
=[( 1+y)+x^2( 1-y)]^2-2( 1+y)x^2( 1-y)-2x^2( 1+y^2)
=[( 1+y)+x^2( 1-y)]^2-(2x)^2
=[( 1+y)+x^2( 1-y)+2x][( 1+y)+x^2( 1-y)-2x]
=(x^2-x^2y+2x+y+ 1)(x^2-x^2y-2x+y+ 1)
=[(x+ 1)^2-y(x^2- 1)][(x- 1)^2-y(x^2- 1)]
=(x+ 1)(x+ 1-xy+y)(x- 1)(x- 1-xy-y)。
You can also see the picture on the right.
2. Verification: For any real number x, y, the value of the following formula will not be 33:
x^5+3x^4y-5x^3y^2- 15x^2y^3+4xy^4+ 12y^5.
Solution: The original formula = (x 5+3x 4y)-(5x 3y 2+15x 2y 3)+(4xy 4+12y 5).
=x^4(x+3y)-5x^2y^2(x+3y)+4y^4(x+3y)
=(x+3y)(x^4-5x^2y^2+4y^4)
=(x+3y)(x^2-4y^2)(x^2-y^2)
=(x+3y)(x+y)(x-y)(x+2y)(x-2y)。
(The factorization process can also be seen in the picture on the right. )
When y=0, the original formula = x 5 is not equal to 33; When y is not equal to 0, x+3y, x+y, x-y, x+2y and x-2y are different from each other, and 33 cannot be divided into products of more than four different factors, so the original proposition holds.
3. The three sides A, B and C of 3.△ ABC have the following relationship: -C 2+A 2+2AB-2BC = 0. Prove that this triangle is an isosceles triangle.
Analysis: This question is essentially factorizing the polynomial on the left side of the relation equal sign.
Prove: ∫-C2+a2+2ab-2bc = 0,
∴(a+c)(a-c)+2b(a-c)=0.
∴(a-c)(a+2b+c)=0.
∵a, B and C are three sides of △ABC,
∴a+2b+c>0.
∴a-c=0，
That is, a = c and △ABC is an isosceles triangle.
4. Factorization-12x2n× y n+18x (n+2) y (n+1)-6xn× y (n-1).
Solution:-12x2n× y n+18x (n+2) y (n+1)-6xn× y (n-1).
=-6x^n×y^(n- 1)(2x^n×y-3x^2y^2+ 1).
You can also see the picture on the right.
Edit the methods used in this competition.
Application of factorial theorem.
For the polynomial f(x)=0, if f(a)=0, then f(x) must contain the factor x-a. 。
For example, if f (x) = x 2+5x+6 and f(-2)=0, it can be determined that x+2 is a factor of x 2+5x+6. (Actually, it is x 2+5x+6 = (x+2) (x+3). )
Substitution method.
Sometimes in factorization, you can choose the same part of the polynomial, replace it with another unknown, then factorize it and finally convert it back. This method is called substitution method.
Note: don't forget to return the RMB after exchange.
For example, when decomposing (x 2+x+1) (x 2+x+2)-12, you can make y = x 2+x, then
The original formula =(y+ 1)(y+2)- 12.
=y^2+3y+2- 12=y^2+3y- 10
=(y+5)(y-2)
=(x^2+x+5)(x^2+x-2)
=(x^2+x+5)(x+2)(x- 1).
You can also see the picture on the right.
(9) Root-seeking method
Let the polynomial f(x)=0 and find its roots as x 1, x2, x3, ... xn, then the polynomial can be decomposed into f (x) = (x-x1) (x-x2) (x-x3) ... (x-xn).
For example, when 2x 4+7x 3-2x 2- 13x+6 is decomposed, let 2x 4+7x 3-2x 2- 13x+6 = 0.
By comprehensive division, the roots of the equation are 0.5, -3, -2, 1.
So 2x4+7x3-2x2-13x+6 = (2x-1) (x+3) (x+2) (x-1).
⑽ image method
Let y=f(x), make the image of function y=f(x), and find the intersection of function image and x axis, x 1, x2, x3, ... Xn, ... xn, then the polynomial can be factorized into f (x) = f (x) = (x-x/kloc-0.
Compared with ⑼ method, it can avoid the complexity of solving equations, but it is not accurate enough.
For example, if you decompose x 3+2x 2-5x-6, you can make y = x 3+2x 2-5x-6.
Make an image, and the intersection with the X axis is -3,-1, 2.
Then x3+2x2-5x-6 = (x+1) (x+3) (x-2).
⑾ Principal component method
First, choose a letter as the main element, then arrange the items from high to low according to the number of letters, and then factorize them.
⑿ Special value method
Substitute 2 or 10 into x, find the number p, decompose the number p into prime factors, properly combine the prime factors, write the combined factors as the sum and difference of 2 or 10, and simplify 2 or 10 into x, thus obtaining factorization.
For example, when x 3+9x 2+23x+ 15 is decomposed, let x=2, then
x^3 +9x^2+23x+ 15 = 8+36+46+ 15 = 105，
105 is decomposed into the product of three prime factors, namely 105 = 3× 5× 7.
Note that the coefficient of the highest term in the polynomial is 1, while 3, 5 and 7 are x+ 1, x+3 and x+5, respectively. When x=2,
Then x 3+9x 2+23x+ 15 may be equal to (x+ 1)(x+3)(x+5), which is true after verification.
[13] undetermined coefficient method
Firstly, the form of factorization factor is judged, then the letter coefficient of the corresponding algebraic expression is set, and the letter coefficient is calculated, thus decomposing polynomial factor.
For example, when x 4-x 3-5x 2-6x-4 is decomposed, the analysis shows that this polynomial has no primary factor, so it can only be decomposed into two quadratic factors.
So let x4-x3-5x2-6x-4 = (x2+ax+b) (x2+CX+d).
=x^4+(a+c)x^3+(ac+b+d)x^2+(ad+bc)x+bd
Therefore, a+c=- 1,
ac+b+d=-5，
ad+bc=-6，
bd=-4。
The solutions are a= 1, b= 1, c=-2 and d =-4.
Then x4-x3-5x2-6x-4 = (x2+x+1) (x2-2x-4).
You can also see the picture on the right.
[14] Double cross multiplication.
Binary multiplication is a kind of factorization, similar to cross multiplication. Use an example to illustrate how to use.
Example: the decomposition factor: x2+5xy+6y2+8x+18y+12.
Analysis: This is a quadratic six-term formula, and you can consider factorization by binary multiplication.
Solution:
x 2y 2
① ② ③
x 3y 6
∴ Original formula = (x+2y+2) (x+3y+6).
Double cross multiplication includes the following steps:
(1) first decompose the quadratic term by cross multiplication, such as x2+5xy+6y 2 = (x+2y) (x+3y) in the cross multiplication diagram (1);
(2) according to the first coefficient of a letter (such as y) to score constant items. For example, 6y2+18y+12 = (2y+2) (3y+6) in the cross multiplication diagram ②;
(3) according to the first coefficient of another letter (such as x), such as cross plot (3). This step cannot be omitted, otherwise it is easy to make mistakes.
Typing is not easy. If you are satisfied, please adopt it.