Mathematical formula method to factorize

1.=[(x+2y)-2z]*[x+2y)+2z]

x+2y-2z)(x+2y+2z)

Brief analysis: The first question is the square difference formula, a 2-b 2=(a+b)(a-b), in this question a is (x+2y), b is 2z.

2.=a^2(a-b)-b^2(a-b)=(a^2-b^2)(a-b)

a-b)(a-b)(a+b)=(a-b)^2(a+b)

Brief analysis: This question is divided into two steps, the first step is to extract the common factor, and pay attention to changing the symbols when extracting. in order to extract the common factor (a-b); Step 2: Is it the formula method, or the square difference formula.

3.=(a-9ab)(a+9ab)

Brief Analysis: Similar to the first question.

4.=(9x^2y-14a^2b)(9x^2y+14a^2b)

Brief analysis: This problem is actually a square difference formula, the positive number of terms in this problem is last, and the negative number of terms is first, so you can put 81x 4y 2

Look at it as a 2, and put it. -196a 2b 2 is regarded as -b 2, therefore, it can be made up a 2-b 2=(a-b)(a+b), this problem also requires the square of familiar numbers, for example, this problem should clearly reflect that the square of 14 is 196.

5.=[2(x+2y)-5(x-2y)][2(x+2y)+5(x-2y)]

2x+4y-5x+10y)(2x+4y+5x-10y)

14y-3x)(7x-6y)

Brief analysis: Don't forget to move items and merge similar items after the breakdown. Complete.

Factorization: Formula method. Items of the same kind that can be merged should be merged.

Formula method: A method of solving a quadratic equation, also refers to applying a formula to calculate a transaction. According to the relationship between factorization and integer multiplication, the coefficients can be directly brought into the root finding formula, which can avoid the formula process and directly obtain the root, and this method of solving the unary quadratic equation is called the formula method.

Steps of the formula method:

1.The equation is general: ax to the 2nd power + bx + c = 0 (a≠0).

2.Determine the discriminant formula and calculate the δ (Greek alphabet, transliterated as Delta). =b to the 2nd power-4ac;

3.If 0 is δ>, the equation has two unequal real roots in the field of real numbers: x=2a [-b (2nd power of b-4ac);

If δ=0, the equation has two equal real roots in the field of real numbers: x1=x2=-2 b;

If δ< 0, the equation has no solution in the real number domain, but there are two conjugate complex roots in the imaginary number field, which is [-b (2 powers of 4ac-b)i in the field of x=2a.

Defactoring: Defactoring a polynomial in a range (e.g., a range of rational numbers, i.e., all terms are rational) into the form of the product of several simplest formulas, this deformation is called factorization, also known as factorization. It has a wide range of applications in mathematical root plotting.

Significance: It is one of the most important identity variations in secondary mathematics, it is widely used in elementary mathematics, and it is a powerful tool for us to solve many mathematical problems. The factorization method is flexible and technical, and learning these methods and skills is not only necessary to master the content of factorization, but also has a very unique role in cultivating students' problem-solving skills and developing students' thinking ability.

Learning it can not only review the four operations of integers, but also lay a good foundation for learning fractions; Learning it well can not only cultivate students' observation, thinking development, and computing ability, but also improve students' ability to comprehensively analyze and solve problems.

Factoring method

1. Mention the common factor method.

If the items of a polynomial have a common factor, you can propose this common factor to reduce the polynomial into the form of the product of two factors, and this method of factoring is called the common factor method.

The common factors that each item contains are called the common factors of each of the polynomials. The common factor can be either a mononomial or a polynomial.

2. Formula method.

If the two sides of the equality sign of the multiplication formula are swapped, the formula used to decompose the factor can be obtained, which is used to defactor some polynomials with a special form, and this method of factoring is called the formula method.

3. Cross multiplication.

The multiplication of the left side of the cross is equal to the coefficient of the quadratic term, the multiplication of the right side is equal to the constant term, and the multiplication of the cross and the addition of the first term are equal to the primary term.

Formula: Divide quadratic terms, divide constant terms, and cross multiply to sum once terms. (Split the two ends and make up the middle).

1) Decompose the quadratic terms by cross multiplication to obtain a cross multiplication graph (with two columns);

2) Decompose the constant term f into two factors and fill in the third column, requiring the first.

2. The sum of the products of the crosses formed by the third column is equal to ey, the first in the original formula.

1. The sum of the products of the crosses formed by the third column is equal to dx in the original formula

3) first score the constant term with a one-letter primary coefficient;

4) Perform another test according to the coefficient of another letter;

5) Add horizontally and multiply vertically.

4. Rotational symmetry method.

When the problem is a rotational symmetry, the rotational symmetry method can be used to decompose.

5. Group decomposition method.

The method of factoring is called the grouping factorization method to decompose the factors that cannot be directly decomposed by the common factor method and the formula factorization method. There are four or more polynomials that can be grouped and decomposed, and there are two forms of general grouping decomposition: binary and trian.

6. Dismantling and adding items.

The method of splitting or filling two (or several terms) of a polynomial that are opposite to each other makes the original formula suitable for the common factor method, the formula method or the group decomposition method, and this method of factoring is called the splitting and complementing method. Note that the deformation must be performed on the principle of equality with the original polynomial.

7. Matching method.

For some polynomials that cannot be factored by the formula method, they can be factored into a perfect square method, and then the squared difference formula can be used to factorize them, and this method of factoring is called the matching method. It is a special case of the method of splitting and supplementing items. It is also important to note that the deformation must be carried out on the principle of equality with the original polynomial.

Definition of formula method: If the two sides of the equality sign of the multiplication formula are swapped, the formula used to decompose the factor can be obtained, which is used to defactor some polynomials with a special form, and the method of defactoring the factor is called the formula method.

Decomposition formula: 1Square Difference Formula:

That is, the difference of squares of two numbers, equal to the product of the sum of these two numbers and the difference between these two numbers.

2.Perfect Square Formula:

That is, the sum of the squares of the two numbers plus (or subtract) 2 times the product of the two numbers, which is equal to the sum (or difference) of the two numbers.

Note: A polynomial that can be factored using a perfectly squared formula must be trinomial, two of which can be written as the sum of the squares of two numbers (or equations) and the other as twice the product of the two numbers (or formulas).

Formula: the first square, the tail square, twice the product **. The same number is added, the different number is subtracted, and the symbol is added before the different number.

The factoring formula method is as follows:

1. Square difference formula: that is, the square difference of two numbers, equal to the product of the sum of these two numbers and the difference between these two numbers.

The characteristics of the formula: the left is the binomial, which is the difference between the perfect squares of two numbers, and the right is the product of the sum and difference of the two numbers.

2. Perfect square formula: that is, the sum of the squares of two numbers plus (or subtract) 2 times the product of these two numbers is equal to the square of the sum (or difference) of these two numbers.

The characteristics of the formula: the left side is the trinomial, the only thing in which the first and last terms are the form of the sum of squares of the two numbers, the middle term is 2 times the product of the two numbers (plus the corresponding symbol), and the right side is the square of the sum (or difference) of the two numbers.

Formula: the first square, the tail square, twice the product **. The same number is added, the different number is subtracted, and the symbol is added before the different number.

Summary of methods and techniques:

1. Square difference formula, in a perfect square formula, the letters a and b in the formula can be replaced by numbers or letters, or by monomials or polynomials.

2. If the items of a polynomial contain common factors, first mention the common factors, and then further decompose them until they can no longer be decomposed.

3. Some calculation problems, although they belong to simple numerical calculation, but according to the general steps, not only the calculation is troublesome, and easy to make mistakes, if you can use the method of factorization, first factorization, and then calculation, you can simplify the operation process in a big way.

Square Difference Formula: (a-b)(a+b)=a2-b 2 Perfect Square Formula:

a+b)^2=a^2+2ab+b^2

a-b)^2=a^2-2ab+b^2

Extract the common factor.

ab+ac=a(b+c)

Cross multiplication.

ax +bx+c=(px+m)(qx+n), where pq=a, pn+qm=b, mn=c

Perfectly squared. ax +bx+c=a(x+b 2a) +c-b 4a, where c-b 4a=0 i.e. c=b 4a

Square difference a -b = (a + b) (a-b).

Sum of squares a +b = (a + bi) (a-bi).

Cube difference a -b = (a-b) (a + ab + b ) sum of cubes.

a³+b³=(a+b)(a²-ab+b²)

Factorization: Formula method. Items of the same kind that can be merged should be merged.

Hello, if we look at a+b as a whole, then use the squared difference formula, which is (a+b) 3 = (a+b+3)(a+b-3), so we do the factoring. It is easier to think of the squared quantity as a whole. Hope it helps.

(a b) squared -9

a b) squared - 3 squared.

a＋b+3）*（a＋b-3）

Factorization: Formula method. Items of the same kind that can be merged should be merged.

Factorization: Formula method. Items of the same kind that can be merged should be merged.

Question 1: The equation is [2(a+b)+5(a-c)][2(a+b)-5(a-c)]=(7a+2b-5c)(-3a+2b+5c).

Question 2: =1 2(a 2-4)=1 2(a+2)(a-2).

Factorization: Formula method. Items of the same kind that can be merged should be merged.

First, use the formula method to find x1 x2 and substitute (x-x1)*(x-x2)=0

Completely flat approach.

Seek the best.