№2
https://www.grandars.ru/student/vysshaya-matematika/formuly-sokrashchennogo-umnozheniya.html
x(x-1)(x-1)-(x-2)(x²+2x+4)=x(x-1)²-(x³-2³)=x(x²-2x+1)-(x³-8)=
=x³-2x²+x-x³+8=-2x²+x+8
№3
1)7m-n+49m²-n²=(7m-n)+(7m-n)(7m+n)=(7m-n)(7m+n+1)
2)4x²-4xy+y²-16=(2x-y)²-16=(2x-y)²-4²=(2x-y+4)(2x-y-4)
3)xy⁴-2y⁴-xy+2y=y⁴(x-2)-y(x-2)=(y⁴-y)(x-2)=y(y³-1)(x-2)
4)9-x²-2xy-y²=9-(x²+2xy+y²)=9-(x+2)²=3²-(x+2)²=(3-x-2)(3+x+2)
№4
1)5x³-5x=0
5x(x²-1)=0
5x(x-1)(x+1)=0
5x=0 ⇒ x₁=0
x-1=0 и x²+1=0 ⇒ x₂ ₃=±1
2)64x³-16x²+x=0
x(64x²-16x+1)=0
x(8x-1)²=0
x₁=0
(8x-1)²=0 ⇒ x₂=1/8
3)x³-3x²-4x+12=0
x²(x-3)-4(x-3)=0
(x²-4)(x-3)=0
(x-2)(x+2)(x-3)=0
x-3=0 ⇒ x₁=3
x-2=0 и х+2=0 ⇒ х₂ ₃=±2
№5
(4⁶-7³)/9=(4³×4³-7³)/9=(16³-7³)/9=(16-7)(16²-7×16+7²)/9
=9×(16²-7×16+7²)÷9=(16²-7×16+7²)
(16²-7×16+7²) ∈ N ⇒ выражение 4⁶-7³ делится нацело на 9
№6
a+b=4 ab=-6 (a-b)²-?
(a-b)²=a²-2ab+b² и (a+b)²=a²+2ab+b²
⇒(a²-2ab+b²)+4ab=a²+2ab+b²
(a-b)²+4ab= (a+b)²
(a-b)²=(a+b)²-4ab=4²-4×(-6)=16+24=40
15 3 3 --- = 3 --- 100 20