Factoring, how to decompose? Is there any trick to it?

To be familiar with the most basic formulas, learn to derive backwards

Factoring Method:

1. If the first term of the polynomial is negative, the negative sign should be extracted first; The "negative" here refers to the "negative sign". If the first term of the polynomial is negative, a negative sign is generally proposed so that the coefficient of the first term in parentheses is positive.

2. If each item of the polynomial contains a common factor, then the common factor is extracted first, and then the factor is further decomposed; Note: When a whole term of a polynomial is a common factor, after proposing the common factor first, do not omit 1 in parentheses; Mention the common factor should be cleaned at once, and the polynomial in each parenthesis can no longer be decomposed.

3. If there is no common factor for each item, then you can try to use formulas and cross multiplication to decompose them;

4. If the above methods cannot be decomposed, try to decompose by grouping, splitting items, and supplementing items.

Formula: first mention the first negative sign, then see if there is a common factor, and then see if you can set the formula, cross multiplication and try it, and the group decomposition should be appropriate.

Factorization mainly includes cross multiplication, undetermined coefficient method, double cross multiplication, symmetric polynomial, rotational symmetric polynomial method, coincidence theorem and other methods.

There is no universally applicable method for finding the common factor decomposition, and the junior high school mathematics textbooks mainly introduce the common factor method, the formula method, and the group decomposition method. In the competition, there are also splitting and adding and subtracting terms, changing the element method, long division, short division, division, etc.

The method of factoring a polynomial is to decompose a polynomial in the form of a range (e.g., a range of real numbers, i.e., all terms are real).

Factorization is one of the most important identity deformations in middle school mathematics, which is widely used in elementary mathematics, and is also widely used in mathematical root plotting and solving one-click clever chain element quadratic equations, which is a powerful tool for solving many mathematical problems.

The factorization method is flexible and skillful. Learning these methods and techniques is not only necessary for mastering the content of factoring, but also has a very unique role in cultivating problem-solving skills and developing thinking skills. 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 calculation skills, but also improve their ability to comprehensively analyze and solve problems.

Factorization is closely related to solving higher-order equations. For unary quadratic equations and unary quadratic equations, junior high school has written relatively fixed and easy methods. It can be mathematically shown that there are also fixed formulas that can be solved for unary cubic equations and unary quadratic equations.

It's just that the formula is too complex and not introduced in the non-specialist field.

For factorization, cubic and quarkey polynomials also have fixed decomposition methods, but they are more complex. For general polynomials of more than five degrees, it has been proved that no fixed factorization method can be found, and there is no fixed solution method for unary equations of more than five degrees. All cubic and cubic more unary polynomials can be factored in the range of real numbers, and all quadratic or more quadratic univariate polynomials can be factored in the range of complex numbers.

1. When using the method of extracting common factors to decompose the factor of a polynomial, the structural characteristics of the polynomial are first observed to determine the common factor of the polynomial. When the common factor of each item of a polynomial is a polynomial, it can be converted into a mononomial by setting auxiliary elements, or the polynomial factor can be regarded as a whole and the common factor can be directly extracted. When the common factor of each item of a polynomial is implicit, the polynomial should be appropriately deformed, or the symbol should be changed until the common factor of the polynomial can be determined.

2. Pay attention to the use of the formula x 2 + (p+q)x+pq=(x+q) (x+p) for factorization

1) The constant term must first be decomposed into the product of two factors, and the algebraic sum of these two factors is equal to the coefficient of the primary term.

2) Multiple attempts to decompose a constant term into a product of two factors that satisfy the requirements, general steps:

List the possible cases in which a constant term is decomposed into the product of two factors;

Try which of the sum of the two factors is exactly equal to the coefficient of the primary term.

3. Decompose the original polynomial into the form of (x+q)(x+p).

4. Group decomposition method.

If we look at the polynomial am+an+bm+bn, there is no common factor in these four terms, so we can't use the extraction common factor method, and then we can't use the formula method to decompose the factor.

If we divide it into two groups (am+an) and (bm+bn), these two groups can be factored separately by extracting the common factor.

Original formula = (am+an) + (bm+bn) = a(m+n) + b(m+n).

Doing this is not called factoring a polynomial, because it doesn't fit the meaning of factoring. However, it is not difficult to see that these two items also have a common factor (m+n), so they can continue to be decomposed, so: the original formula = (am+an) + (bm+bn) = a(m+n) + b(m+n) (a+b)

This method of factoring using grouping is called grouping decomposition. As you can see from the above example, if the terms of a polynomial are grouped and the common factor is extracted, then the polynomial can be factored by grouping factorization.

Hello, you can use the common factor method, and then see if you can use the formula method to factor it, mainly using the square difference formula and the perfect square formula.

1.If the factor has a minus sign, the minus sign 2 should be extracted firstIf the factor has a common factor, extract the common factor first3 If there is no common factor for each item, then try to use the formula Cross Multiplication to break down the factor.

4.If the above methods do not work, use the methods of grouping, splitting items, and supplementing items to decompose.

1. Extracting the common factor method: the most basic and simple method is to extract the same letter contained in each mononomial in the polynomial and turn it into a multiplicative form.

2. Square difference method: If two items are subtracted and each term is squared, then it can be decomposed by the square difference formula.

3. Perfect square method: If the polynomial contains three terms and satisfies the form of perfect square, it can be decomposed by the perfect square formula.

4. Cross multiplication: The most classic method, and the most commonly used, decompose two of them, add them by cross multiplication, and if they are equal to the third term, you can decompose the factor.

5. Group decomposition method: In view of the situation that the number of terms is relatively large, it is relatively complex, so it is necessary to group according to the characteristics of the formula, and then to combine different groups, which requires sufficient observation.

Factoring in a range of real numbers.

Is there a problem, I can solve it, you see.

There are 2 types of factorization.

1.Mention the public factor.

Coefficient: Find the greatest common factor.

Letters: 1Find the same letters.

2.Find the one with the smallest exponent of the same letter.

Such as 3a -9ab

Extract the greatest common divisor 3

The same letter is A

Solution=3a(a-3b).

The result is to be.

The form of an integer product.

2.Formula method.

Square Difference Formula = Only two terms, two squares subtracted.

a²-b²=(a+b)(a-b)

For example, (xy) -1=(xy+1)(xy-1)1 can be seen as 1

The perfect square formula has only three terms.

Form a completely flat way.

a+b)²=(a²+2ab+b²)

2ab = twice the product of two numbers.

Such as x + 4xy + 4y

x²+4xy+(2y)²

x²+x*2y*2+(2y)²

Conforms to the perfect squared formula.

x+2y)²

Step 1 Is there a common factor?

Whether it's a formula or not.

Decompose until you can't decompose.

Notice that the answers come to you are.

The form of an integer product.

There is a common factor to mention the common factor first, and then use the formula method to calculate, you are in the second year of junior high school, which is very simple.

(1) Mention the common factor method: MA+MB+MC=M(A+B+C) (2) Formula method: A

Square difference formula: a2-b2=(a+b) (a-b)bPerfect Square Formula:

a2 2ab+b2=(a b)2 When factoring a factor, first consider whether there is a common factor, and if there is a common factor, first propose the common factor, and then consider the factorization.

What is written in the math book, do you take a closer look?