1.
xyz(x？ +y？ +z？ )-x？ y？ -Really? z？ -z？ x？
=x^4yz+xyz(y？ +z？ )-y？ z？ -x？ (y？ +z？ )
=yz(x^4-y？ z？ )+x(yz-x？ )(y？ +z？ )
=yz(x？ +yz)(x？ -yz)-x(x？ -yz)(y？ +z？ )
=(x？ -yz)(x？ yz+y？ z？ -xy？ -xz？ )
=(x？ -yz)[y？ (z？ -xy)-xz(z？ -xy)]
=(x？ -yz)(y？ -xz)(z？ -xy)
2. It is known that x+y-z is a compound X? +axy+by？ A factor -5x+y+6, find the values of A and B, and decompose the factor.
x？ +axy+by？ -5x+y+6=(x-3)(x-2)+axy+y+by？ .
According to cross multiplication, the factorization form of this formula should be (x+my-3)(x+ny-2).
①m= 1，z=3。 At this time, the corresponding coefficients are n=b, 1+n=a, -2-3n= 1, n=- 1, a=0, and b=- 1.
x？ -Really? -5x+y+6=(x+y-3)(x-y-2)
②n= 1，z=2。 At this time, the corresponding coefficients are m=b, 1+m=a, -3-2m= 1, m=-2, a=- 1, and b=-2.
x？ -xy-2y？ -5x+y+6=(x+y-2)(x-2y-3)
3. it is known that a+b+c= 1, a? +b？ +c？ =2，a？ +b？ +c？ =3, find the value of abc.
Shit (a+b+c)? =a？ +b？ +c？ +2(ab+bc+ca)
Get 1? =2+2(ab+bc+ca)，ab+bc+ca=- 1/2 .
Factorization formula
Answer? +b？ +c？ -3abc=(a+b+c)(a？ +b？ +c？ -ab-bc-ca)
3-3abc= 1*(2+ 1/2)，abc= 1/6 .
4.a, B, and C are three sides of a triangle, which satisfies a? - 16b？ -c？ +6ab+ 10bc=0, which proves that a+c=2b.
Answer? - 16b？ -c？ +6ab+65438+ 00 BC
=(a+c)(a-c)+b(6a+ 10b)- 16b？
=(a+c-2b)(a+c+8b) (cross multiplication)
Because a, b and c are three sides of a triangle, A+C+8B >: 0,
Therefore, a+c-2b=0, that is, a+c=2b. Get a license.
5. known x? +x+ 1=0, find the value of 2003 power of x+2002 power of x+1.
x？ +x+ 1=0 Multiply both sides by x to get x? +x？ +x=0, then x? = 1。
Similar ones are x 4 = x and x 5 = x? ,x^6=x^3= 1，
That is, x (3k+m) = x m (k is a natural number and m=0 or 1 or 2).
So x 2003+x 2002+ 1.
=x^(3*667+2)+x^(3*667+ 1)+ 1
=x？ +x+ 1=0