Who will be in junior high school math cross multiplication

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Question Decomposition factor: 2x2+3xy+y2-x-2y-3

Misconception] original = (2x+y)(x+y)-(x+2y)-3

Positive solution 1] Split item grouping.

Original = (2x2+2xy+2x)+(y2+xy+y)+(3x-3y-3).

2x（x+y+1）+y（x+y+1）-3（x+y+1）（x+y+1）（2x+y-3）。

Positive solution 2] Organize the polynomial into a quadratic trinomial with respect to the letter x and decompose it by cross multiplication.

Original = 2x2+(3y-1)x+(y2-2y-3)2x2+(3y-1)x+(y-3)(y+1)(x+y+1)(2x+y-3).

Positive solution 3] Pending coefficient method:

2x2+3xy+y2-x-2y-3=（2x+y）（x+y）-x-2y-3 （1）

Let the original formula = [(2x+y)+m][(x+y)+n](2x+y)(x+y)+m(x+y)+n(2x+y)+mn(2x+y)(x+y)+(m+2n)x+(m+n)y+mn (2) compare the coefficients of the corresponding terms in (1) and (2) to obtain.

Original = (2x+y-3)(x+y+1).

Error Cause Analysis and Problem Solving Guidance].

Misunderstanding] only one step has been done, and it can't be done anymore.

Three solutions are provided, and methods such as group decomposition and cross multiplication are flexibly applied to solve the problem.

Practice Questions] Factoring:

Practice question answers].

1.（x+3y-2）（x-2y+3）

2.（x+y-2）（x+2y+1）

You write as follows:

For example, x - 4x +3=0

The coefficient of x --- constant term.

The multiplication of the crosses becomes 1*-1 +1*-3 =-4, and where -4 is the coefficient of the primary term.

You can use the matching method to summarize the cross multiplication by yourself.

Many students have learned cross multiplication in mathematics, so what does cross multiplication refer to? How should we use cross multiplication?

The method of cross decomposition method is simply as follows: the left side of the cross is multiplied equals the quadratic term, the right side multiplied is equal to the constant term, and the cross multiplication and then added is equal to the coefficient of the primary term. In fact, it is to use multiplication formula operations to factorize.

Cross multiplication is a method of factorization. For the one-element quadratic formula, it is more convenient when the integer coefficient is used. Of course, it is better to have a quadratic coefficient of 1. When you decompose a one-time formula, it's easier to think that a constant term is an integer, but a fraction can also be.

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

In fact, it is to use the inverse operation of the multiplication formula (x+a)(x+b)=x + (a+b)x+ab to factorize.

The method of cross multiplication: formula: divide into quadratic terms, divide constant terms, and cross multiply to sum the first terms.

AhhI've been tired of junior high school for a long time...Let's tell you this, for example: x 2+7x+15

To make sure that the product of the multiplication on the left side of the cross is equal to the coefficient of the quadratic term (in this case, it is multiplied to get 1), and then the constant term of the right multiplication (in this problem, it is multiplied to get 15), and then at the same time, ensure that the coefficient of the first term of the addition is equal to 7! In this way, only 3 and 5 satisfy the conditions (i.e. 3x5=15, 3+5=8), so finally multiplying the cross (i.e., the lower left corner by the upper right corner, the other side is the same) gives (x+3) (x+5)=0 to get the two roots, respectively -3 and -5! I hope you understand...It's very useful...It saves a lot of time when solving equations.

For example, the square of x plus 4x equals 3. First, divide the square of x into two x products (vertically), then divide 3 into 1 and 3 (both vertically), and finally write the cross multiplication formula. Such as:

x+1)(x+3)=0 bet, if you are not sure, you can count it again.

For example, the square of x plus 5x plus 6, the coefficient before x square can be divided into 1*1, and 6 can be divided into 2*3 or 1*6, because the coefficient of x is 5, 2+3=5, so the original formula can be decomposed into (x+2)*(x+3).Got it?

Cross multiplication - The method of factoring a quadratic trinomial formula by drawing the decomposition coefficient of the cross line is called cross multiplication. Cross multiplication can factor certain quadratic trinomials. For an integer of the form ax 2+bx+c=(a1x+c1)(a2x+c2), the key to the method is to decompose the quadratic coefficients a into the product a1·a2 of the two factors a1 and a2, decompose the constant term c into the product c1·c2 of the two factors c1 and c2, and make a1c2+a2c1 exactly the coefficient b of the primary term, then the result can be directly written:

ax^2+bx+c=(a1x+c1）(a2x+c2）。When using this method to decompose a factor, 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 often takes several tests, and it is important to pay attention to the symbols of each coefficient.

The basic formula: x 2+(p+q) +pq=( +p)( q).

General steps of factorization.

1) If each item of the polynomial has a common factor, the common factor should be extracted first;

2) If the terms of the polynomial do not have a common factor, consider whether they can be decomposed by the formula method;

3) For the factorization of quadratic trinomials, the cross-multiplication method can be considered.

4) For polynomials with more than three terms, the group decomposition method should generally be considered.

When factoring, it is necessary to choose which method to use based on the form and characteristics of the problem. The above four methods are related to each other, and it is not a type of polynomial that can only be factored in one way, and it is necessary to learn to analyze specific problems.

When we do the questions, we can refer to the following formula:

First extract the common factor, and then consider using the formula;

Cross multiplication test to divide the group appropriately;

The four methods are tried repeatedly, and the final one must be multiplication.

Although cross multiplication is more difficult to learn, once we learn it, using it to solve problems will bring us a lot of convenience.

1. The method of cross multiplication: the left side of the cross is equal to the quadratic term coefficient, the right side is equal to the constant term, and the cross multiplication and then the addition is equal to the primary term coefficient.

2. The usefulness of cross multiplication: (1) Use cross multiplication to decompose factors. (2) Use cross multiplication to solve a quadratic equation.

3. Advantages of cross multiplication: the speed of solving problems by cross multiplication is relatively fast, which can save time, and the amount of calculation is not large, and it is not easy to make mistakes.

4. Defects of cross multiplication: 1. Some problems are relatively simple to solve by cross multiplication, but not every problem is simple to solve by cross multiplication. 2. Cross multiplication is only applicable to quadratic trinomial type problems. 3. Cross multiplication is more difficult to learn.

Definition of cross multiplication: The method of factoring a quadratic trinomial formula by using a cross to decompose coefficients is called cross multiplication.

The so-called cross multiplication method is to use the inverse operation of the multiplication formula (x+a)(x+b)=x 2+(a+b)x+ab to factorize. For example, factor x 2+7x+12.

The constant 12 of the above equation can be decomposed into 3*4, and 3+4 is exactly equal to the coefficient of the primary term 7, so.

The above equation can be decomposed as: x 2 + 7x + 12 = (x + 3) (x + 4).

For example, if you decompose the factor: a 2 + 2 a-15, the constant -15 of the above equation can be decomposed into 5 * (-3).

And 5+(-3) is exactly equal to the coefficient of the primary term 2, so a 2+2a-15=(a+5)(a-3)It's as simple as that. Give it a try!

Factoring by cross multiplication: 1, x 2-x-12 2、x^2+x-20

x 2+(p+q)x+pq=(x+p)(x+q) divides the quadratic term x 2 into the constant term pq, and then cross-multiplies them so that their sum is the primary term.

For example: 1: x 2+5x+6=x 2+(2+3)x+2x3=(x+2)(x+3).

2.2x 2-4x-6=(2x+2)(x-3) divides the quadratic term 2x 2 into the constant term -6 into 2x(-3) and cross-multiplies by 2x 2

x -32xx(-3)+xx2=-4x

This gives 2x 2-4x-6=(2x+2)(x-3).

Cross multiplication is a quadratic equation.

For example: x 2-3x+2=0

x-1)(x-2)=0

It is the factor that decomposes the constant factor, the sum of the two factors as the coefficient of the primary term, and the product of the two factors as the constant.

The new textbook no longer requires cross multiplication. However, if you use it when doing the questions, the teacher will not deduct points.

It's not easy to write symbols here, I'll show you the pictures in my lesson plan, I hope it helps.

First, decompose the coefficient of the quadratic term, which is written in the upper left corner and the lower left corner of the cross line, and then decompose the constant term, divided.

Don't write in the upper right and lower right corners of the crosshairs, then cross and multiply to find the algebraic sum so that it is equal to the coefficient of the primary term.

Decompose quadratic coefficients (take only positive factors, because taking negative factors will result the same as positive factors!)

Let's take an example.

For example, ax 2+bx+c=0

Then using the cross multiplication method is to divide +a into two numbers and multiply them, such as d*f, and +c into two numbers, such as g*h

then d gf h

dh*fg=+b then the above equation can be reduced to (dx+g)(fx+h)=0

Positive and negative signs should also be taken into account, such as 2x 2-3x-2=0, which can be reduced to 2 1 2*(-2)+1*1=-3

So it becomes (2x+1)(x-2)=0

Decompose the quadratic coefficients first, and then decompose the constant terms so that the cross-multiplication is equal to the primary coefficients.

Solving a quadratic equation?