How binary cubic equations are factored

x^3-6x^2y+11xy^2-6y^3

x^3-6x^2y+9xy^2) +2xy^2-6y^3)

x(x^2-6xy+9y^2) +2y^2(x-3y)

x(x-3y)^2 + 2y^2(x-3y)

x-3y)x^2 - 3xy + 2y^2) *x-3y)

x-y)(x-2y)(x-3y)

1. First of all, it is necessary to clarify factorization.

The range of the number field. Cubic polynomials.

It may or may not be reducible in the field of rational numbers (reducible means factorable). It must be reducible in both the real and complex domains. If the factorization is in the field of real numbers or complex numbers, you can use the Cardan formula to directly find the root for factorization.

It is discussed below, its factorization within the field of rational numbers.

2. Then, take advantage of Eisenstein.

The discriminant method determines whether it is negotiable. If it is irreducible, then it cannot be factored within the field of rational numbers; If it is reducible, then it has at least one root in the field of rational numbers.

3. Finally, under the premise that the rational number domain is reducible, the rational root theorem of the integer coefficient polynomial is used to judge the rational root. Using the obtained rational roots, the result of factorization can be written very quickly. At this point, factorization is complete.

Unary cubic equation solving:

1. Shengjin formula method:

Cubic equations are widely used. Solve a cubic equation with a root number.

Although there is the famous Caldan formula.

There is a corresponding discriminant method, but the use of Caldan's formula to solve the problem is more complex and lacks intuitiveness. Fan Shengjin derives a set of general formulas for a more concise form of a cubic equation expressed directly in a, b, c, and d.

A new root-seeking formula, the Shengjin formula, and a new discriminant method, the Shengjin discriminant method, was established.

2. Shengjin Judgment Method:

When a=b=0, the equation has a triple real root.

When δ=b2 4ac>0, the equation has a real root and a pair of conjugate imaginary roots.

When δ=b2 4ac=0, the equation has three real roots, one of which is a double root.

When δ=b2 4ac<0, the equation has three unequal real roots.

3. Sheng Jin's theorem:

When b=0 and c=0, Shengjin's equation 1 is meaningless; When a=0, Shengjin's equation 3 is meaningless; When a 0, Shengjin's equation 4 is meaningless; When t<1 or t>1, Shengjin's equation 4 is meaningless.

The factorization of binary cubic equations can be obtained by extracting the formula method, and the specific solution is that x 3-2x 2-x+2=0 can be decomposed into (x-2)(x-1)(x+1)=0.

The specific solution process is as follows:

First of all, its constant term is 2, so its factor has , -1;Then substitute a random number such that x 3-2x 2-x+2=0;For example, if you bring in 2, the result is 2 3-2*2 2-2+2=0, and the original formula holds; So it is proved that there is definitely one of the factors that is (x-2); Then substitute the original formula (x-2), i.e., :

x^3-2x^2-x+2

x^2(x-2)-(x-2)

x-2)(x^2-1）

x-2)(x-1)(x+1)

It can be obtained by extracting the formula method. Reduces a polynomial to several integers in a range (e.g., decomposition within a range of real numbers, i.e., all terms are real).

The product of the form, this sub-deformation of the formula is called the factorization of this polynomial.

It's also called factoring the polynomial.

Factorization is one of the most important identity deformations in middle school mathematics, and it is widely used in elementary mathematics to find roots and make graphs and solve unary quadratic equations.

It is also widely used and is a powerful tool for solving many mathematical problems.

The factorization method is flexible and the masking technique is strong. 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 the whole formula, but also lay a good foundation for learning the town and learning the division formula; 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.

Solution: 1. Put |λe-a|If they are equal, then the equal parts of the stool ruler belt are put forward (the primary factor), and the remaining part is a quadratic polynomial, which can definitely be factored.

2. Put |λe-a|When one of the two elements that are not present in a row (or a column) is zeroed, a common factor often appears, and the rest is a quadratic polynomial.

3. Decomposition of factors by root test method.

Properties: When a is the upper triangular matrix (or the lower triangular moment jujube reed array), <> where <> is

is an element on the main diagonal. For second-order squares, the eigenpolynomial can be expressed as .

In general, if <>

then <>

In addition: 1) the eigenpolynomial does not change under the basis change: if there is an invertible square matrix c so that.

then <>

2) <> on any two phalanxes

Yes <>

In general, if a is <>

matrix, b is <>

Matrix (set trapped<> then <>

3) Gloria-Hamiltomy theorem:

Unary quadratic equations can be solved by factorization.

The general form of a quadratic equation:

The general form of a quadratic equation is ax +bx+c=0, where a, b, and c are known numbers, and a≠0.

Methods of factorization:

For the unary quadratic equation ax +bx+c=0, it can be solved by factorization. The specific methods are as follows:

1.Divide both sides of the equation by a to get x +b'x+c'a=0, where b'=b/a，c'=c/a。

2.Replace x + b'x+c'A is expressed as (x+m)(x+n), where m and n are the coefficients to be determined.

3.Combine (x+m)(x+n) to get x +(m+n)x+mn=0.

4.Compare the coefficients and get m+n=b'，mn=c'a, i.e., m and n are c'a, and their sum is b'。

5.Find the values of m and n, substitute (x+m)(x+n)=0, and get the solution of the equation.

Expand your knowledge:

1.When the discriminant formula b -4ac>0 of a quadratic equation is used, the equation has two unequal real roots; When b -4ac=0, the equation has two equal real roots; When b -4ac < 0, the equation does not have a real root in the middle of the equation, but has two conjugate complex roots.

2.The method of factorization can also be used to solve other types of equations, such as unary cubic equations, binary quadratic equations, etc.

3.The method of factorization can also be used to simplify the operation of polynomials, such as multiplication, division, simplification of polynomials, etc.

Expressing the equation x +5x+6=0 in the form of (x+m)(x+n) gives x + (m + n) x + mn = 0. Compare the sum coefficient to obtain m+n=5 and mn=6. Since m and n are two factors of 6, and their sum is 5, m=2 and n=3.

Therefore, the solution of the equation is x=-2 or x=-3.

In summary, a quadratic equation can be solved by factorization. The method of factorization can be applied to the operation of other types of equations and polynomials, and is one of the basic methods in algebra.

The deformation of a polynomial into the product of several integers is called factorization of the polynomial, also known as factorization of the polynomial.

The matching method is a panacea, but the cross multiplication method is the fastest.

extraction of common factor method;

group decomposition method;

Cross multiplication---a(x-p)(x-q)=0;

The matching method ---a(x-m) +n=0;

Formula method: x= (2a).

Cross multiplication.

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 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.

For example: a x +ax-42

First, let's look at the first number, which is a, which means that it is obtained by multiplying two a's, then we deduce (a +a + and then we look at the second term, +a is the result obtained by merging similar terms, so it is inferred to be binomial.

If you look at the last term, it is -42, and -42 is -6 7 or 6 -7, which can also be broken down into -21 2 or 21 -2.

First of all, 21 and 2, regardless of whether they are positive or negative, cannot be 1 after arbitrary addition or subtraction, but can only be -19 or 19, so the latter is excluded.

Then, determine whether it is -7 6 or 7 -6.

a -7) (a +6) = a x -ax-42 (omitted from the calculation).

The result obtained does not match the original result, and the original formula +a becomes -a.

Again: (a +7) (a +(6)) = a x +ax-42

correct, so a x +ax-42 is decomposed into (ax+7) (ax-6), which is the popular cross multiplication factor.

Formula methodFormula method, that is, the use of formulas to decompose factors.

The formula generally has.

1、a²-b²=（a+b）（a-b）

2、a²±2ab+b²=（a±b）²

Binary cubic equations.

Such as Gao Qing to eliminate the factor.

For example, x 3-6x 2y+11xy 2-6y 3=0 can be decomposed into (x-y)(x-2y)(x-3y)=0, how to decompose?

There are rewards for writing a total of 3 answers.

A holmium 呗钬唋u

Let's talk about following to become the 196th fan.

The factorization of binary cubic equations can be obtained by extracting the formula method, and the specific solution is that x 3-2x 2-x+2=0 can be decomposed into (x-2)(x-1)(x+1)=0.

The specific solution process is as follows:

Let's start with its constant term.

is 2, so its factor has , -1;Then substitute a random number such that x 3-2x 2-x+2=0;For example, if you bring in 2, the result is 2 3-2*2 2-2+2=0, and the original formula holds; So it is proved that there is definitely one of the factors that is (x-2); Then substitute the original formula (x-2), i.e., :

x^3-2x^2-x+2

x^2(x-2)-(x-2)

x-2)(x^2-1）

x-2)(x-1)(x+1)

Factorization.

Deform the equation to zero on one side and quadratic trinomials on the other.

Decompose into the form of the product of two linear factors, so that the two primary factors are equal to zero respectively, and obtain two unary linear equations.

The roots obtained by solving these two unary equations are the two roots of the original equation. This solution is a quadratic equation.

The method is called factorization.

Example: The following equation is solved using the factorization method:

1)(x+3)(x-6)=-8(2)2x2+3x=0

3) 6x2+5x-50=0 (elective) (4) x2-2(+) x+4=0 (elective).

x2-3x-10 = 0 (quadratic trinomials on the left and zero on the right).

x-5)(x+2)=0 (decomposition factor on the left side of the equation.

x-5=0 or x+2=0 (converted into two unary linear equations).

x1=5, x2=-2 is the solution of the original equation.

2) Solution: 2x2+3x=0

x(2x+3)=0 (using the common factor method.

Factor the left side of the equation).

x=0 or 2x+3=0 (converted into two unary equations).

x1=0, x2=- is the solution of the original equation.

Note: It is easy to lose the solution of x=0 when doing this kind of problem, and you should remember that there are two solutions to a quadratic equation.

3) Solution: 6x2+5x-50=0

2x-5) (3x+10) = 0 (pay special attention to the symbols when multiplying the cross to factor, so as not to make mistakes).

2x-5=0 or 3x+10=0

x1=,x2=- is the solution of the original equation.

4) Solution: x2-2(+)x+4=0 (4 can be decomposed into 2·2, this problem can be factored).

x-2)(x-2)=0

x1=2, x2=2 is the solution of the original equation.

Some equations, the highest number of unknowns is 2, containing two unknowns, this kind of equation is called a binary quadratic equation.

Binary bionic equations generally appear in the form of a system of equations, which is called a bivariary bipolar equation. When solving a system of binary quadratic equations, one of the unknowns must be eliminated and the equation must be solved by turning the equation into a binary equation.

Some binary quadratic equations give the sum of two unknowns and the product of these two unknowns, we can think of it as two solutions of a two-dimensional equation, and solve this one-dimensional quadratic equation to obtain two sets of solutions of a two-dimensional equation.

I hope I can help you with your doubts.