How to mention the common factor, how to mention the common factor

If the terms of a polynomial have a common factor, you can put the common factor out of parentheses and write the polynomial as a factor product. When the coefficients are integers, the coefficients of the common factor should be taken as the greatest common divisor of the coefficients; The letters are the same as each letter, and the index of each letter is the lowest; Take the same polynomial, and the number of polynomials is the lowest.

The same factor contained in each item of the polynomial is called the common factor of each item of the polynomial, and the method of determining the common factor:

1. The coefficients of the common factor are the greatest common divisor of the coefficients of the polynomial.

2. The letters are the same letters contained in each item of the polynomial.

3. The exponent of the same letter takes the smallest of the terms, that is, the lowest power.

Specific method: when all coefficients are integers, the coefficients of the common factor should be taken as the greatest common divisor of each coefficient; The letters are the same as each letter, and the index of each letter is the lowest; Take the same polynomial, and the number of polynomials is the lowest.

If the first term of the polynomial is negative, it is common to put a "-" sign so that the coefficient of the first term in parentheses becomes positive. When the "-" sign is proposed, each item of the polynomial must be changed.

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Obviously, it also requires a certain amount of skill to mention the common factoring method.

Let's look at another example problem: (y-x) 2+y-x =(y-x) 2+(y-x)=(y-x+1)(y-x).

Methods for determining the common factor:

General steps for determining common factors.

1) If the first coefficient of the polynomial is negative, the sign of the common factor "-" should be used"Extract.

2) Take the coefficients of the polynomial's coefficients as the coefficients of the common factor.

3) The product of the lowest power of the same letter (or factor) contained in each item of the polynomial is used as the factor of the common factor.

The above step is not absolute, step (1) can be omitted when the first term is positive.

It can be obtained from the question. x(x-2)+(x-2)*1=0 At this time, add parentheses to the following x-2, which does not affect the result of the whole question, and adding any number multiplied by 1 is equal to itself, so the next is to mention the common factor, so there is what you wrote above.

x－2）（x＋1）＝0

Think about it slowly and see if this is the case, it's a good thing to use your brain more, and I hope your math is getting better and better.

x（x－2）＋x－2＝0

They all have a common cause test, (x-2).

x*(x-2)+1*(x-2)=0

Extraction common cause test (x-2) * (x-1) = 0

The same is taken out and the rest is added or subtracted.

Let x-2=a, then the original formula becomes xa+a=0, now you put a out, try, become a(x+1)=0, and then replace a=x-2 back, that is, (x-2)(x+1)=0

The important thing is to look at (x-2) as a whole

The original formula is x*(x-2)+1*(x-2)=0

The common factor is (x-2) and is extracted.

So (x-2)*(x+1)=0

This is equivalent to x(x 2) 1(x 2) 0

The common factor is (x 2).

So there are (x 2) (x 1) 0

Your solution seems to be correct, but you're not multiplying the cross in the common factor method. You can now think of x(x 2) x 2 0 as x(x 2) 1 (x 2) 0, and if you mention a (x 2), isn't that (x 2) (x 1) 0?

The method you are told is a bit unanswered, but your method is actually correct. It's just not the most common method of cross multiplication.

λ³-4λ+λ4＝0

Let's put forward the first two items first.

This shows that the previous part has the same common factor -4 as the following one, and this common factor is proposed.

The result is -2 or 2 or 1.

As shown in the figure below, the first three and two items are combined, and the two and four are combined.

1. When using the common factor method to decompose the factor, it is generally carried out in two steps:

1. Mention the common factor. The product of the lowest power of the same letter or factor in each item is put forward as a common factor; When the coefficients are integers, the greatest common divisor of them should also be proposed as the coefficients of the common factor; When the first sign of the polynomial is negative, a negative sign is also proposed.

2. Use the common factor to divide the bright and do not remove each roll of the polynomial, and write the algebraic sum of the obtained quotient as another factor, and write the form of the product with the common factor.

2. Mention the common factor method: Generally, if the items of the polynomial have a common factor, you can put the common factor outside the brackets, and write the polynomial in the form of the product of the factor, and this method of decomposing the factor is called the common factor method.

1. When using the common factor method to decompose the factor, it is generally carried out in two steps:

1. Mention the public because of Li Cong. The product of the lowest power of the same letter or factor in each item is proposed as the common factor; When the coefficients are integers, the greatest common divisor of them should also be proposed as the coefficients of the common factor; When the first sign of the polynomial is negative, a negative sign is also proposed.

2. Remove each term of the polynomial with the common factor, and write the algebraic sum of the obtained quotient as another factor, and the common factor as a product.

2. Mention the common factor method: Generally, if the items of the polynomial have a common factor, you can put the common factor outside the parentheses, and write the polynomial in the form of the product of the factor, and this method of decomposing the factor is called the common factor method.

Generally, ifPolynomialsThere is a common sail missing the factor, you can put this common factor outside the parentheses, and write the polynomial in the form of the product of the factor, this wayDecompose the factorThe method is calledMention the common factor method

Specific method: when all coefficients are integers, the coefficients of the common factor should be taken as the greatest common divisor of each coefficient;

The letters are the same letters, and the exponents of each letter are the lowest number of states; Take the same polynomial, and the number of polynomials is the lowest.

Extraction of common cause collapse method.

It is a basic method of factorization. If the items of a polynomial have a common factor, you can extract this common factor as a factor of the polynomial, and put the formula after the extraction of the common factor in parentheses as another factor.

Extracting the common factor is the multiplicative distributive property.

The inverse of , in its simplest form, is: ma+mb+mc=m(a+b+c).

The common factor method is the first method of factorization, and it is also the first method that should be considered when getting a factorization problem, so the common factor is the most basic and important method How to learn to factor the common factor method? 1. Clarify the basis for mentioning the common factor The basis for mentioning the common factor is the multiplicative distributive law: ma+mb+mc=m(a+b+c) Second, the steps of decomposing the common factor method:

1. The first thing to mention the common factor is to find out whether there is a common factor for each item through observation 2. If each item of the polynomial has a common factor, then the greatest common divisor of each coefficients and the lowest power of the letter of each item are required, and the product of the two is used as the common factor of the polynomial to be decomposed 3. Write each item as the product of the common factor and another monomial 4. Write the final result Example 1: Decompose the factor: 32a b -16a b +24a b Analysis: The common factor of this multiple is a monomial, It is necessary to consider the solution in terms of coefficients and letters

Original formula = 8a b 4a -8a b 2ab + 8a b 3b = 8a b (4a -2ab + 3b) Third, several problems to pay attention to when decomposing factors by the common factor method 1. To overcome the "missing term" When one of the terms in the polynomial is extracted as a common factor, the position of this item should be "1", which cannot be omitted or omitted Example 2, decomposition factor: 3x -7xy + x solution: original formula = x 3x - x 7y + x 1 = x (3x -7y+1) In order to prevent this error, write x as x 1, so that it can be seen that after proposing a factor x, the other factor is 1 Students can use the following smooth slip to help memorize:

What is a common factor? All in common to each item, a certain item is all proposed, leaving '1' to guard the house" 2, to deal with the first coefficient is "-" when the first coefficient of the polynomial is negative, generally the "-" sign is mentioned outside the parentheses, so that the first coefficient of the polynomial in the parentheses is positive, so that the deformation is conducive to us to observe how the latter decomposes the factor But it should be noted that when the "-" sign is proposed, the symbols of each item of the polynomial should be changed Example 3, decomposition factor: -2a b + 3a +4a solution:

2a b+3a +4a=-a 2b-a (-3a)-a (-4a)=-a (2b-3a-4a)3, when the common factor is a polynomial, you need to pay attention to the sign change If the polynomial items are only one negative sign apart, then after deformation, such a formula becomes the common factor of the polynomial

Take the same numbers and letters, and then put a parenthesis in isolation.