x^5+3x^4y-5x^3y^2+4xy^4+ 12y^5
Solution: The original formula = (x 5+3x 4y)-(5x 3y 2+15x 2y 3)+(4xy 4+12y 5).
=x^4(x+3y)-5x^2y^2(x+3y)+4y^4(x+3y)
=(x+3y)(x^4-5x^2y^2+4y^4)
=(x+3y)(x^2-4y^2)(x^2-y^2)
=(x+3y)(x+y)(x-y)(x+2y)(x-2y)
Three principles of attention
1 decompose thoroughly.
The final result is only parentheses.
The first term coefficient of the polynomial in the final result is positive.
Inductive method:
1, common factor method
The common factor of each term is called the common factor of each term of this polynomial.
If every term of a polynomial has a common factor, we can put forward this common factor, so that the polynomial can be transformed into the product of two factors. This method of decomposing factors is called the improved common factor method.
Specific methods: when all the coefficients are integers, the coefficients of the common factor formula should take the greatest common divisor of all the coefficients; The letter takes the same letter of each item, and the index of each letter takes the smallest number; Take the same polynomial with the lowest degree. When the coefficient of each term has a fraction, the denominator of the common factor coefficient is the least common multiple of the denominator of each fraction, and the numerator is the greatest common factor (the greatest common factor) of each fractional numerator.
If the first term of a polynomial is negative, a "-"sign is usually put forward to make the coefficient of the first term in brackets become positive. When the "-"sign is put forward, the terms of the polynomial should be changed.
Formula: find the right common factor and clean it up once; The whole family moved out and left 1 to look after the house; The negative sign should be changed, and the deformation depends on parity.
For example:-am+BM+cm =-(a-b-c) m;
a(x-y)+b(y-x)= a(x-y)-b(x-y)=(a-b)(x-y)
Note: Replacing 2a+ 1/2 with 2(a+ 1/4) is not a common factor.
2. Formula method.
If the multiplication formula is reversed, some polynomials can be factorized. This method is called formula method.
Square difference formula: (a+b) (a-b) = a 2-b 2.
On the other hand, a 2-b 2 = (a+b) (a-b)
Complete square formula: (a+b) 2 = a 2+2ab+b 2.
On the contrary, it is a 2+2ab+b 2 = (a+b) 2.
(a-b)^2=a^2-2ab+b^2
a^2-2ab+b^2=(a-b)^2
Note: Polynomials that can be decomposed by the complete square formula must be trinomial, two of which can be written as the sum of squares of two numbers (or formulas), and the other is twice the product of these two numbers (or formulas).
Two formulas: ax 2+bx+c = a (x-(-b+√ (B2-4ac))/2a) (x-(-b-√ (B2-4ac))/2a).
Cubic sum formula: a 3+b 3 = (a+b) (a 2-ab+b 2)
Cubic difference formula: a 3-b 3 = (a-b) (a 2+ab+b 2)
Complete cubic formula: a 3 3a 2b+3ab 2 b 3 = (a b) 3.
Formula: A3+B3+C3-3abc = (A+B+C) (A2+B2+C2-AB-BC-CA)
For example: a 2+4ab+4b 2 = (a+2b) 2.
3. Grouping decomposition method.
4. Number method. [x^2+(a+b)x+ab=(x+a)(x+b]
5. Combinatorial decomposition.
6. Cross multiplication.
Cross multiplication can decompose some quadratic trinomials. The key of this method is to decompose the quadratic coefficient A into two factors A 1, the product of A2 a 1, A2 and the constant term C into two factors c 1, and multiply c2 C 1 by c2, so that C 1, C2+A2C/kl.
When using this method to decompose factors, we should pay attention to observation and try to understand that it is essentially the inverse process of binomial multiplication. When the first coefficient is not 1, it often needs to be tested many times, so be sure to pay attention to the sign of each coefficient.
Basic formula: x 2+(p+q) χ+PQ = (χ+p) (χ+q). The so-called cross multiplication is factorization by using the inverse operation of the multiplication formula (x+a) (x+b) = x 2+(a+b) x+ab. For example:
Another example is the factorization factor: a 2+2A-15, and the constant of the above formula-15 can be decomposed into 5×(-3). And 5+(-3) is exactly equal to the linear coefficient of 2. So A 2+2A- 15 = (A+5) (A-3)
Cross multiplication explains:
x^2-3x+2
As follows:
x - 1
╳
x -2
Left x times x = x 2
Right-1 times -2=2
Middle-1 times x+(-2) times x (diagonal) =-3x.
The x+(- 1) above is multiplied by the x+(-2) below.
It is equal to (x- 1)*(x-2).
x^2-3x+2=(x- 1)*(x-2)
7. Double cross multiplication.
8. Matching method.
9. Demolition method.
10, substitution method.
1 1, long division.
12, addition and subtraction.
13, root method.
14, mirror method.
15, principal component method.
16, undetermined coefficient method.
17, special value method.
18, factorial theorem method.