What is group decomposition and cross multiplication? Formula? Wield?

Cross Phase: The left side of the cross is multiplied equally equals the quadratic coefficient, the right side is multiplied equals the constant term, and the cross multiplication and then the addition are equal to the primary term coefficient. Cross multiplication can factor certain quadratic trinomials.

The key to this method is to decompose the quadratic coefficient a into the product of two factors a1,a2, decompose the constant term c into the product of two factors c1,c2 times c2, and make a1c2+a2c1 exactly a term b, then the result can be directly written: ax 2 + bx + c = (a1x+c1)(a2x+c2).

When factoring a method, it is important to observe, try, and realize that it is essentially the inverse of binomial multiplication.

When the first coefficient is not 1, it is often required.

Process. It is necessary to test several times, and it is important to pay attention to the symbols of each coefficient. Group decomposition method: group items appropriately, first make the decomposition factor grouped, and then make the decomposition factor between the groups

Parenthesis: The parenthesis is preceded by a "+" sign, and the items in the parentheses are invariant symbols; The parentheses are preceded by a "-" sign, and the symbols in the parentheses are changed. When the number of terms in the polynomial is large, the polynomial can be reasonably grouped to achieve the purpose of smooth decomposition.

Of course, it may be necessary to combine other divisions, and the grouping method is not necessarily unique.

Help you! Hope.

The factoring of cross multiplication is explained as follows:

The cross decomposition method can be used to factor the quadratic trinomial and unary quadratic formulas, not necessarily in the range of integers. For an integer like ax2+bx+c=(a1x+c1)(a2x+c2), the key to this method is to decompose the quadratic coefficient a into the product of the two factors a1a2, and the constant term c into the product of the two factors c1c2, so that a1c2+a2c1 is exactly equal to the coefficient b of the primary term.

Then you can write the result directly: ax2+bx+c=(a1x+c1)(a2x+c2). When using this method to decompose factors, it is important to observe, try, and realize that it is essentially the inverse process of binomial multiplication.

When the first coefficient is not 1, it often takes several tests, and it is important to pay attention to the symbols of each coefficient. Basic formula: x2+(p+q)x+pq=(x+p)(x+q).

Example. 1) Example 1: x2-x-56;

Analysis: Because 7x+(-8x)=-x;

Solution: Original = (x+7) (x-8).

2) Example 2: x2-10x+16;

Analysis: Because -2x+(-8x)=-10x;

Solution: Original formula = (x-2) (x-8).

Cross multiplication

The method of cross decomposition method is simply as follows: the left side of the cross is multiplied equals the quadratic term coefficient, the right side is multiplied equals the constant term, and the cross multiplication and then addition equals the primary term coefficient is actually the use of multiplication formula operation to factorize.

Cross multiplication is one of the fourteen methods in factorization, and the other thirteen are the following methods: the common factor method, the formula method, the double cross multiplication method, the rotational symmetry method, the addition method, the matching method, the factoring theorem method, the commutation method, the comprehensive division method, the principal element method, the special value method, the undetermined coefficient method, and the quadratic polynomial.

The quadratic coefficients and constant terms in a quadratic equation are split into the product of two numbers, so that the sum of the two products of the four numbers crossing the two is exactly equal to the coefficient of the primary term, which reduces the equation to the product of two linear equations. For a quadratic equation x 2 + 4 x + 3 = 0

Because 1 = 1 * 1, 3 = 1 * 3Therefore, the equation is equivalent to (x+1)(x+3)=0, and the equation 2x 2+x-1=0 is equivalent to (2x-1)(x+1)=0+

It is to decompose the number in front and the number in the back into the form of multiplying two numbers, and then cross multiply and add to get the middle number, so that it is true, for example: 2x 2+3x-5=0 is rewritten as (x-1)(2x+5)=0 and then solve the answer.

If I'm not mistaken, the Shanghai version of the textbook is factorized in Unit 8 of the first volume. Cross multiplication, formula method and matching method are in the book for the second semester of the second semester of junior high school.

Factorization: x +2x+1=(x+1) ; x^+6x+9=(x+3)^

Cross multiplication This is a cross multiplication exercise, you still have to do a little more) matching method: it is a completely flat method, which is inseparable from the factorization method.

Formula method: x=b -4ac divided by 2a (the formula method is a universal method, the golden key to solving a quadratic equation).

3] Cross multiplication.

It is the basic method of doing competition questions, and it will be easy to master this by doing the usual questions. Note: It's not difficult.

The key to this method is to decompose the quadratic term coefficient a into the product a1 a2 of the two factors a1 and a2, and decompose the constant term c into the product c1 c2 of the two factors c1 and c2, and make a1c2+a2c1 exactly the first term b, then it can be directly written as the result.

Example 3: Factor 2x 2-7x+3.

Analysis: First decompose the quadratic term coefficients and write them in the upper left and lower left corners of the cross line, then decompose the constant terms and write them in the upper right and lower right corners of the cross line, and then multiply them to find the algebraic sum so that it is equal to the primary term coefficient.

Decompose quadratic coefficients (take only positive factors):

Decomposition Constant Term:

The method of drawing crosses is used to represent the following four situations:

7 After observation, the fourth case is correct, and this is because after cross-multiplication, the sum of the two algebras is exactly equal to the coefficient of the first term 7

Solution = (x-3)(2x-1)

Summary: For the quadratic trinomial ax 2+bx+c(a≠0), if the quadratic term coefficient a can be decomposed into the product of two factors, i.e., a=a1a2, and the constant term c can be decomposed into the product of two factors, i.e., c=c1c2, a1, a2, c1, c2, arrange as follows:

a1 c1 ╳a2 c2

a1c2+a2c1

Multiply according to the diagonal crossing, and then add to get a1c2+a2c1, if it is exactly equal to the primary term coefficient b of the quadratic trinomial ax2+bx+c, that is, a1c2+a2c1=b, then the quadratic trinomial can be decomposed into the product of two factors a1x+c1 and a2x+c2, i.e.

ax2+bx+c=(a1x+c1)(a2x+c2).

This method should be more experimental, more done, and more practiced. It can include the first two methods.

Factorization example: a 2-1 = (a + 1) (a - 1) cross multiplication: a 2 + 5a + 4 = (a + 1) (a + 4).

Formula: -b+-root number b 2-4ac 2a matching method: a 2+2a-6=0 a 2+2a+1=7 (a+1) 2=7

It's all in the books.