Упростите выражение с объяснением: 1) (a²-3)³-(a-2)(a²+4)(a+2)
2)(b²-3)³-(b²+3)(b^4-3b²+9)
3)(m²-1)(m^4+m²+1)-(m²-1)³
4)(x²-2)(x^4+2x²+4)-(x³-1)²

ответ и Объяснение:

Нужно знать формулы сокращённого умножения:

a) a²-b² = (a-b)·(a+b);

b) (a-b)² = a²-2·a·b+b²;

c) a³+b³ = (a+b)·(a²-a·b+b²);

d) a³-b³ = (a-b)·(a²+a·b+b²);

e) (a-b)³ = a³-3·a²·b+3·a·b²-b³.

1) (a²-3)³-(a-2)·(a²+4)·(a+2) = [e)] = a⁶-3·a⁴·3+3·a²·3²-3³-(a-2)·(a+2)·(a²+4) = [a)] =

= a⁶-9·a⁴+27·a²-27-(a²-4)·(a²+4) = [a)] = a⁶-9·a⁴+27·a²-27-(a⁴-16) =

= a⁶-9·a⁴+27·a²-27-a⁴+16 = a⁶-10·a⁴+27·a²-11;

2) (b²-3)³-(b²+3)(b⁴-3·b²+9) = [e), c)] = b⁶-3·b⁴·3+3·b²·3²-3³-(b⁶+27) =

= b⁶-9·b⁴+27·b²-27-b⁶-27 = -9·b⁴+27·b²-54;

3) (m²-1)(m⁴+m²+1)-(m²-1)³ = [d), e)] = m⁶-1-(m⁶-3·m⁴·1+3·m²·1²-1³) =

= m⁶-1-(m⁶-3·m⁴+3·m²-1) = m⁶-1-m⁶+3·m⁴-3·m²+1 = 3·m⁴-3·m²;

4) (x²-2)·(x⁴+2·x²+4)-(x³-1)² = [d), b)] = x⁶-8-(x⁶-2·x³+1) =

= x⁶-8-x⁶+2·x³-1 = 2·x³-9.