Elementary 1 polynomial cross multiplication factorization 5

Turning a polynomial into the product of several integers, a sub-deformation like this is called factoring the polynomial, also known as factoring the polynomial.

The relationship between factorization and integer multiplication: reversible deformation (some polynomials can be factored, some cannot be factored).

Steps:1If you can mention the common factor, you should mention the common factor.

2.If you can't mention the common factor, it depends on whether it can be formulated or not.

3.The factoring must be factored to the point where it can no longer be factored.

Mention the common factor method: 1coefficient to find the greatest common divisor.

2.Letters are found to be the same letters, and the exponent is the lowest power.

For the rest, you can read the whole content once, and it's not good for me to learn factorization at the beginning, so it's good to practice more.

For example, x 3-3x +2x

x(x^2-3x+2)

x 2-3x+2 = as follows: x -1

x -2 x x to the left x = x 2

Right -1 times -2 = 2

Middle -1 times x + -2 times x (diagonal) = -3x

The top x+(-1)*the bottom x+(-2)] is equal to (x-1)*(x-2).

x^2-3x+2=（x-1）*（x-2）

This is something I typed out little by little, and I have to give points.

You can't find a second one as detailed!!

Give it points!!

(x+p)(x+q)=x2+(p+q)x+pq, so to decompose such a formula, you must first look at the constant.

See which two numbers a constant can be broken down into; Then add the two numbers together to see if they are equal to the coefficient of x.

For example: x2 5x 4

2 2=4≠-5

Original = (x-1)(x-4).

Hope you understand.

The cross multiplication for factorization is as follows:

The cross multiplication method of the factorization method is that the left side of the cross is multiplied to equal the quadratic coefficient, the right side is multiplied to be equal to the constant term, and the cross multiplication and then added is equal to the primary term coefficient.

Cross multiplication is one of the fourteen methods in factorization, and the method of cross multiplication is simply as follows: 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 addition are equal to the primary term. In fact, it is to use multiplication formula operations to factor up and recall.

Cross multiplication can be used to factorize quadratic trinomials (not necessarily in the range of integers).

For an integer like ax +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 a1 and a2, and the constant term c into the product of the two factors c1 and c2, so that a1c2+a2c1 is exactly equal to the coefficient b of the primary term. Then you can write the result directly: ax +bx+c=(a1x+c1)(a2x+c2).

When using this method to solve the factor, we should pay attention to observe, try, and realize that its essence is 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）。

The method of cross decomposition 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 addition is equal to the primary term.

For a polynomial of the shape ax2+bx+c, δ=b2-4ac can be used to determine whether it can be factored using the cross decomposition method. When the δ is a perfectly squared number, the polynomial can be cross-multiplied in an integer range.

1. The cross decomposition method can be used to decompose the factor of the quadratic trinomial (not necessarily within the range of integers) 2. For integers like ax2+bx+c=(a1x+c1)(a2x+c2), the key to this method is to decompose the quadratic coefficient a into the product of two factors a1 and a2, so that a1c2+a2c1 is exactly equal to the coefficient b of the primary term.

Defactoring is a very important and basic calculation skill in mathematics, which is widely used in solving polynomials, algebraic formulas and other mathematical fields. Among them, the cross multiplication method (cross-multiplication to check the sum of the positive method) is a common method for factoring indication.

Cross multiplication can be used to solve a quadratic polynomial and a solution to a quadratic equation separately. The main idea of this method is to multiply each term in a polynomial by each term of the other, and then add or subtract the result of the multiplication to finally get the factor of the polynomial.

Here are the steps to implement cross multiplication:

1.The polynomials are arranged according to certain rules, where the first two terms are the quadratic parts of the equation, which are formed by the form a"Quadratic terms"x + bx + c in a, b, c composition.

2.According to the order of the polynomials, multiply a and c to get ac, split b into two numbers p and q, satisfy p+q = b and pq=ac, and find the two numbers p and q.

3.The equation is in the form of "left multiplication, right multiplication, and left multiplication", and fill in the found p and q to form the pattern of (ax + p) (x + q).

4.Comparing the original equation with the obtained new equation, it can be seen that the two equations are equivalent, so the new equation is converted into the original equation to obtain the decomposed form of the polynomial.

Through the first three steps, the staggered addition and product conditions are extracted from the multiplication on the left (ax + p) and the multiplication on the right (x + q) to obtain the final decomposition formula.

It is important to note that cross multiplication is suitable for factoring one-dimensional quadratic polynomials, and requires that the coefficients of the polynomial be integers. This method is inefficient when solving complex or higher-order polynomials and requires the help of other methods or tools.

In conclusion, cross multiplication is a very practical method of factoring, which is of great significance when learning polynomial, algebraic and other mathematical skills.

1 All crosses are multiplied.

The basic formula: x 2+(a+b)x+ab=(x+a)(x+b).

This one is very practical, but it is not easy to use.

When it is not possible to decompose it by other methods, you can use the lower cross multiplication.

Example: x 2 + 5 x + 6

First of all, it is observed that there are quadratic terms, primary terms, and constant terms, which can be multiplied by crosses.

The coefficient of the primary term is 1So it can be written as 1*1

The constant term is 6It can be written as 1*6, 2*3, -1*-6, -2*-3 (decimals are not recommended).

Then arrange 1- like this

The positions of the following columns can be reversed, as long as the product of these two numbers is a constant term).

Then multiply diagonally, 1*2=2, 1*3=3Add the product again. 2+3=5, which is the same as the coefficient of the primary term (it may not be equal, in this case another attempt should be made), so it can be written as (x+2) (x+3).

At this point, just come sideways).

I'll write a few more formulas, and the landlord will figure it out for himself.

x^2-x-2=(x-2)(x+1)

2x^2+5x-12=(2x-3)(x+4)

In fact, the most important thing is to use it yourself, the above methods can actually be used together, and practice is always better than teaching others.

By the way. If the b 2-4ac of an equation is less than 0, the formula cannot be decomposed in any way (in the range of real numbers, b is the coefficient of the first term, a is the coefficient of the quadratic term, and c is the constant term).

These methods are generally applicable when the highest order is secondary!

4y^2+6y+4

2（2y^2+3y+2)

With the extraction common factor method, this is the end of the factorization.

2y 2+3y+2 is not good for cross multiplication, no matter what, it can't match the middle 3

2y 2+3y+2 is not easy to use the root method, because the discriminant formula of its root =3 2-4x2x2=9-16=

7<0, no real root, no solution.