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.
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.
(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.
⑶ Grouping decomposition method
Group decomposition is a simple method to solve equations. Let's learn this knowledge.
There are four or more terms in an equation that can be grouped, and there are two forms of general grouping decomposition: dichotomy and trisection.
For example:
ax+ay+bx+by
=a(x+y)+b(x+y)
=(a+b)(x+y)
We put ax and ay in a group, bx and by in a group, and matched each other by multiplication and division and distribution, which immediately solved the difficulty.
Similarly, this problem can be done.
ax+ay+bx+by
=x(a+b)+y(a+b)
=(a+b)(x+y)
A few examples:
1.5ax+5bx+3ay+3by
Solution: =5x(a+b)+3y(a+b)
=(5x+3y)(a+b)
Note: Different coefficients can be decomposed into groups. As mentioned above, 5ax and 5bx are regarded as a whole, and 3ay and 3by are regarded as a whole, which can be easily solved by using the multiplication and distribution law.
2.x3-x2+x- 1
Solution: =(x3-x2)+(x- 1)
=2x(x- 1)+(x- 1)
=(x- 1)(x2+ 1)
Using dichotomy, 2x is proposed by common factor method, and then it is easy to solve.
3.x2-x-y2-y
Solution: =(x2-y2)-(x+y)
=(x+y)(x-y)-(x+y)
=(x+y)(x-y+ 1)
Use dichotomy, then use the formula a2-b2=(a+b)(a-b), and then skillfully solve it.
(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)。
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)。
[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 and ad+bc=m, then kx 2+MX+n = (ax+b) (CX+d).
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.
Point reference material
There are examples above.