If every term of a polynomial contains a common factor, then this common factor can be put forward, so that the polynomial can be transformed into the product of two factors.

Example 1, factorization factor x-2x-x (Huai' an senior high school entrance examination in 2003)

x -2x -x=x(x -2x- 1)

2. Application of formula method

Because factorization and algebraic expression multiplication have reciprocal relationship, if the multiplication formula is reversed, it can be used to decompose some polynomials.

Example 2, factorization factor a +4ab+4b (2003 Nantong senior high school entrance examination in 2003)

Solution: a +4ab+4b =(a+2b)

3. Grouping decomposition method

Factorizing the polynomial am+an+bm+bn, we can first divide the first two terms into a group and propose the common factor A, then divide the last two terms into a group and propose the common factor B, so as to get a(m+n)+b(m+n), and we can also propose the common factor M+N, so as to get (a+b) (m+).

Example 3. Decomposition factor m +5n-mn-5m

Solution: m +5n-mn-5m= m -5m -mn+5n.

= (m -5m )+(-mn+5n)

=m(m-5)-n(m-5)

=(m-5)(m-n)

4. Cross multiplication

For a polynomial in the form of mx +px+q, if a×b=m, c×d=q and ac+bd=p, the polynomial can be factorized into (ax+d)(bx+c).

Example 4, factorization factor 7x-19x-6

Analysis: 1 -3

7 2

2-2 1=- 19

Solution: 7x-19x-6=(7x+2)(x-3)

5. Matching method

For those polynomials that cannot be formulated, some can use it to make a completely flat way, and then factorize it with the square difference formula.

Example 5, Factorization Factor x +3x-40

Solution x +3x-40=x +3x+() -() -40

=(x+ ) -()

=(x++)(x++)

=(x+8)(x-5)

The basis of solving equations

1, move terms and symbols: move some terms in the equation from one side of the equation to the other side of the previous symbol, and add, subtract, multiply and divide, change, multiply and divide;

2. The basic properties of the equation

Property 1: Add (or subtract) the same number or the same algebraic expression on both sides of the equation at the same time, and the result is still an equation. Represented by letters: if a=b, c is a number or an algebraic expression.

( 1)a+c=b+c

(2)a-c=b-c

Property 2: Both sides of the equation are multiplied or divided by the same number that is not 0 at the same time, and the result is still an equation.

Represented by letters: if a=b, c is a number or an algebraic expression (not 0). Then:

A×c=b×c or a/c=b/c

Property 3: If a=b, then b=a (symmetry of the equation).

Property 4: If a=b and b=c, then a=c (transitivity of the equation).