The common factor of each polynomial is called the common factor of this polynomial. Usually, some terms or terms of some polynomials have a common factor, and we can put forward this common factor, so that polynomials can be transformed into the product of two factors or multiple factors. This method of decomposing factors is called the improved common factor method.

Second, the formula method. ?

Some polynomial factors can be decomposed by reversing the multiplication formula. This method is called formula method.

Third, the group decomposition method.

Grouping decomposition is a method to decompose more complex polynomials. Polynomials that can be grouped often have four or more terms, and are generally grouped into groups of two or three. It is often used that some terms in polynomials have a common formula or can be simplified by formulas after being merged separately.

Fourth, cross multiplication.

Cross left multiplication equals quadratic coefficient, right multiplication equals constant term, and cross multiplication plus equals linear coefficient. In fact, it is to use the multiplication formula (x+a)(x+b)=x? +(a+b)x+ab for factorization.

Five, double cross multiplication.

Break it down like an axe? +bxy+cy？ Quadratic sextuples of +dx+ey+f on draft paper, A is decomposed into mn product as one column, C into pq product as the second column, and F into jk product as the third column. If mq+np=b, pk+qj=e, mk+nj=d, that is, column 1, 2, 2.

Then the original formula =(mx+py+j)(nx+qy+k). Also known as long cross multiplication.

Extended data:

I. Decomposition Theorem of Polynomials:

Any polynomial with degree not less than 1 in F[x] can be decomposed into the product of irreducible polynomials on f, and the method of decomposition is unique except the order of factors and constant factors.

When F is a complex field C, according to the basic theorem of algebra, it can be proved that the irreducible polynomials in C[x] are all linear. Therefore, every polynomial with complex coefficients can be decomposed into the product of the first factor.

When f is a real number field R, the irreducible polynomial in R[x] is linear or quadratic because the imaginary roots of real coefficient polynomials appear in pairs, that is, the * * * yoke of the imaginary roots is still the roots.

Therefore, every polynomial with real coefficients can be decomposed into the products of some irreducible polynomials of first and second degree. The necessary and sufficient condition for real coefficient quadratic polynomial αx2+bx+с to be irreducible is its discriminant B2-4α с.

When f is a rational number field q, the situation is much more complicated. It is difficult to judge whether rational coefficient polynomials are irreducible. Using primitive polynomial theory, the decomposition problem of rational coefficient polynomial can be transformed into the decomposition problem of integer coefficient polynomial.

If the coefficients of an integer coefficient polynomial are coprime, it is called primitive. Every rational coefficient polynomial can be expressed as the product of a rational number and a primitive polynomial.

Second, the relevant conclusions of factor decomposition:

1, factorization is closely related to solving higher-order equations. Linear equation and quadratic equation have relatively fixed and easy-to-master methods in junior high school. Mathematically, it can be proved that there are also fixed formulas for solving cubic equations and quartic equations. Just because the formula is too complicated, it will not be introduced in non-professional fields.

For factorization factors, cubic polynomials and quartic polynomials also have fixed decomposition methods, but they are more complicated. It has been proved that there is no fixed factorization method for general polynomials of degree five or above, and there is no fixed solution for univariate equations of degree five or above.

2. All univariate polynomials with more than three degrees can be decomposed in the real number range, and all univariate polynomials with more than two degrees can be decomposed in the complex number range. This may seem a little incredible. For example, x4+ 1, which is a univariate quartic polynomial, seems impossible to factorize.

But its degree is higher than 3, so it must be factorized. It can also be decomposed by undetermined coefficient method, but the formula after decomposition is irregular. (This is because, from the basic theorem of algebra, we can know that an n-degree unary always has n roots, that is, an n-degree unary can always be decomposed into the product of n linear factors.

And there is another theorem: if the imaginary roots of real coefficient polynomials are paired with each other, the quadratic real coefficient factor can be obtained by multiplying the first factor corresponding to the imaginary roots of each pair of yokes. This conclusion holds. )

3. Although there is no fixed method for factorization, there is a fixed method for finding the common factor of two polynomials. Factorization is often used to extract common factors. Finding the common factor can be obtained by tossing and turning division.

The standard skill of division over and over is quite difficult for middle school students, but sometimes the number of polynomials that middle school students have to deal with is not too high, so it is ok to use polynomial division repeatedly. Although it is stupid, it can effectively solve the problem of finding common factors.

4. Factorization is very difficult, and junior high school only touches a very simple part of factorization.

References:

Baidu encyclopedia-factor decomposition