Question

1 вариант 1. Разложите многочлен на множители. 1) 3ха2 - 3хс2; 2) 64p2 -1; 3) - 5m2 +5; 4) 162ay2 - 2a; 5) 12x2 + 12xy+3y2; 6) 27x3 + 3xy2 - x?y; 7) m2 - n2 + m - n; 8) аб - а4 + а2 - 1; 9) 9x2 - 12x + 4 - y2; 10) c2 - x2 - 2xy - y2; 3 11) x2 - 2xy + y2 - 8x + 8y; 12) 4a2 + 4ab + b2 + 12a + 6b; 13) m2(m - 3) - 2m(m - 3) + (m - 3); 14) a2 + 4ab - 3a2b - 6ab2 + 4b2. ​

Answer 1

$1) 3xa^{2} -3xc^{2} =3x(a^2-c^2)=3x(a-c)(a+c)$

$2) 64p2 -1=(8p-1)(8p+1)$

$3)-5m^2+5=5(1-m^2)=5(1-m)(1+m)$

$4) 162ay^2 - 2a=2a(81y^2-1)=2a(9y-1)(9y+1)$

$5)12x^{2} +12xy+3y^2=3(2x+y)^2=3(2x+y)(2x+y)$

$6) 27x3 + 3xy^2 - x^2y=x(27x^{2} +3y^2-y)$

$7) m^2 - n^2 + m - n= (m^2 - n^2) + (m - n)=$

   $=(m+n)( m - n) +( m - n)= ( m - n) ( m + n+1)$

$8) ab -a^4 + a^2 - 1=ab -a^4 + a^2 - 1$

$9) 9x^2 - 12x + 4 - y^2= (9x^2 - 12x + 4 )- y^2= (3x-2)^2 - y^2=$

  $=(3x-2)^2 - y^2= (3x-2-y)(3x-2+y)$

$10) c^2 - x^2 - 2xy - y^2=c^2 -( x^2 + 2xy +y^2)=c^2-(x+y)^2=$

 $=(c-x-y)(c+x+y)$

$11) x^2 - 2xy + y^2 - 8x + 8y=(x^2 - 2xy + y^2) - (8x - 8y)=$

$=(x- y)^2 - 8(x - y)=(x-y)(x-y-8)$

$12) 4a^2 + 4ab + b^2 + 12a + 6b=(4a^2 + 4ab + b^2) + (12a + 6b)=$

$=(2a+b)^2 + 6(2a +b)=(2a+b)(2a+b+6)$

$13) m^2(m - 3) - 2m(m - 3) + (m - 3)=(m-3)(m^2-2m+1)=(m-3)(m-1)^2$

$14) a^2 + 4ab - 3a^2b - 6ab^2 + 4b^2=( a^2 + 4ab +4b^2)- (3a^2b +6ab^2)=$

$=( a+2b)^2- 3ab(a +2b)=(a +2b)(a +2b-3ab)$

Answer 2

1. 3ха^2 - 3хс^2 = 3х(a^2 - с^2) = 3х(a + с)(a - с)

2. 64p^2 - 1 = (8p)^2 - 1 = (8p + 1)(8p - 1)

3. -5m^2 + 5 = -5(m^2 - 1) = -5(m + 1)(m - 1)

4. 162ay^2 - 2a = 2a(81y^2 - 1) = 2a(9y + 1)(9y - 1)

5. 12x^2 + 12xy + 3y^2 = 3(4x^2 + 4xy + y^2) = 3(2x + y)^2

6. 27x^3 + 3xy^2 - xy = x(27x^2 + 3y^2 - y) = x(9x + y)(3x - y)

7. m^2 - n^2 + m - n = (m - n)(m + n) + (m - n) = (m - n)(m + n + 1)

8. аб - а^4 + а^2 - 1 = а^2(а - 1)(а + 1) - (а^4 - 1) = а^2(а - 1)(а + 1) - (а^2 - 1)(а^2 + 1) = (а^2 - 1)(а^2 - а + 1)

9. 9x^2 - 12x + 4 - y^2 = (3x - 2)^2 - y^2 = (3x - 2 - y)(3x - 2 + y)

10. c^2 - x^2 - 2xy - y^2 = (c - x)^2 - (x + y)^2 = (c - x - x - y)(c - x + x + y) = (c - 2x - y)(c + y)

11. x^2 - 2xy + y^2 - 8x + 8y = (x - y)^2 - 8(x - y) = (x - y)(x - y - 8)

12. 4a^2 + 4ab + b^2 + 12a + 6b = (2a + b)^2 + 6(2a + b) = (2a + b)(2a + b + 6)

13. m^2(m - 3) - 2m(m - 3) + (m - 3) = (m - 3)(m^2 - 2m + 1) = (m - 3)(m - 1)^2

14. a^2 + 4ab - 3a^2b - 6ab^2 + 4b^2 = (a^2 - 3a^2b) + (4ab - 6ab^2) + 4b^2 = a^2(1 - 3b) + 2ab(2 - 3b) + 4b^2 = (a^2 + 2ab + 4b^2)(1 - 3b)