Question

(x+1)(x-1)(x⁴+x²+1)-(x²-1)³=

Answer 1

Ответ:

-x²(x²-1)

Объяснение:

(x+1)(x-1)(x⁴+x²+1)-(x²-1)³= (x²-1)x⁴+x²+1)-(x²-1)(x²-1)²=(x²-1)((x⁴+x²+1)-(x²-1)²)=(x²-1)(x⁴+x²+1-x⁴-2x²-1)=(x²-1)(-x²)= -x²(x²-1)

Answer 2

To simplify the expression $(x+1)(x-1)(x^4+x^2+1)-(x^2-1)^3$, you can follow these steps:

1. Expand each part of the expression:

   • Expand $(x+1)(x-1)$: $(x+1)(x-1) = x(x-1) + 1(x-1) = x^2 - x + x - 1 = x^2 - 1$

   • Expand $(x^2-1)^3$: $(x^2-1)^3 = (x^2-1)(x^2-1)(x^2-1)$ Use the binomial formula $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$: $(x^2-1)^3 = (x^2)^3 - 3(x^2)^2(1) + 3(x^2)(1^2) - 1^3$ Simplify further: $(x^2-1)^3 = x^6 - 3x^4 + 3x^2 - 1$

2. Substitute the expanded expressions back into the original expression:

   $(x+1)(x-1)(x^4+x^2+1) - (x^2-1)^3$ $(x^2 - 1)(x^4 + x^2 + 1) - (x^6 - 3x^4 + 3x^2 - 1)$

3. Distribute and simplify:

   $x^6 + x^4 + x^2 - x^4 - x^2 - 1 - (x^6 - 3x^4 + 3x^2 - 1)$

4. Combine like terms:

   $x^6 - x^6 + x^4 + 3x^4 - x^4 + x^2 - 3x^2 - 1 + 1$

5. Simplify further:

   $x^4 - x^2$

So, $(x+1)(x-1)(x^4+x^2+1) - (x^2-1)^3$ simplifies to $x^4 - x^2$.