2. Common factorization methods:
(1) common factor extraction method:
(2) Using formula method: variance formula:;
Complete square formula:
(3) Cross multiplication:
(4) Grouping decomposition method: After properly grouping polynomial terms, common factors can be extracted or decomposed by formulas.
(5) Using the root formula method:
If the two roots of are, there are:
3, the general steps of factorization:
(1) If each term of the polynomial has a common factor, then the common factor should be raised first;
(2) Put forward common factor formula or common factor formula, and then consider whether formula or cross multiplication can be used;
(3) For the quadratic trinomial, we should first try to cross-multiply and decompose, if not, then use the radical method.
(4) Finally, consider using group decomposition method.
Knowledge points of junior high school Olympic mathematics: 2 (1) common divisor and maximum common divisor.
The common divisor of several numbers is called the common divisor of these numbers; The largest one is called the greatest common divisor of these numbers.
For example, 4 is the greatest common divisor of 12 and 16, which can be written as: (12, 16)=4.
(2) Common multiple and minimum common multiple
The common multiple of several numbers is called the common multiple of these numbers; The smallest one is called the least common multiple of these numbers.
For example, 36 is the least common multiple of 12 and 18, and it is recorded as [12, 18]=36.
(3) the relationship between the greatest common divisor and the least common multiple
If two natural numbers are represented by a and b,
1, then the relationship between the greatest common divisor and the least common multiple of these two natural numbers is:
(a，b)×[a，b]=a×b .
(mostly used to find the least common multiple)
2 、( a，b) ≤ a，b ≤ [a，b]
3.[a, b] is a multiple of (a, b), and (a, b) is a divisor of [a, b].
4.(a, b) is the divisor of a+b and a-b, and it is also the divisor of (a, b)+[a, b] and (a, b)-[a, b].
(4) There are many ways to find the greatest common divisor, mainly including: short division, prime factor decomposition and division by turns.
For example:
1, (short division) 30, 60, 75 divided by a number can be divided exactly. What is the maximum quantity?
Solution:
(30,60,75)=5×3= 15
The maximum number is 15.
2. What is the greatest common divisor of 100 1 and 308?
Solution:1001= 7×113 (commonly used in this qualitative decomposition), 308=7× 1 1×4.
So the greatest common divisor is 7× 1 1=77.
In this method, the numbers are first decomposed qualitatively, and then their "product of all prime factors" is taken as the greatest common divisor.
3. Find the greatest common divisor of 48 1 1 and 198 1 by division.
Solution: ∫ 4811= 2×1981+849,
198 1=2×849+283,
849=3×283,
∴(48 1 1, 198 1)=283。
Supplementary note: If you need the greatest common divisor of three or more numbers, you can first find the greatest common divisor of any two of them, then find the greatest common divisor of this common divisor and another number, and so on until you get the final result.
(5) divisor formula
The divisor of a composite number is equal to the number (i.e. exponent) of each prime factor in its prime factor decomposition formula plus the product of 1.
For example, find the divisor of 240.
Solution: ∫240 = 24×3 1×5 1,
The divisor of 240 is
(4+ 1)×( 1+ 1)×( 1+ 1)=20,
240 has 20 divisors.