Which of these expressions is equivalent to z^8–1?
(z^4+1)(z^2–1)(z–1)(z–1)
(z^4–1)(z^2–1)(z–1)(z–1)
(z^4+1)(z^2+1)(z+1)(z–1)
(z^4+1)(z^2+1)(z–1)(z–1)
I Think it is either A or C, is one of these correct?
let a = z^4
(a^2-1) = (a-1)(a+1)
= (z^4-1)(z^4+1)
=(z^2-1)(z^2+1)(z^4+1)
=(z^4+1)(z^2+1)(z+1)(z-1)
Yes, one of the options A and C is correct. Let's simplify each option step by step to determine which one is equivalent to z^8–1.
Option A: (z^4+1)(z^2–1)(z–1)(z–1)
Expanding the brackets:
= (z^4+1)(z^2–1)(z^2–2z+1)
Simplifying further:
= (z^6 – z^4 – z^2 + 1)(z^2–2z+1)
= z^8 – 2z^7 + z^6 – z^6 + 2z^5 – z^4 + z^4 – 2z^3 + z^2 – z^2 + 2z – 1
After combining like terms:
= z^8 – 2z^7 + 2z^5 – 2z^3 + 2z – 1
Option C: (z^4+1)(z^2+1)(z+1)(z–1)
Expanding the brackets:
= (z^4+1)(z^2+1)(z^2 + z – z + 1)
Simplifying further:
= (z^6 + z^4 + z^2 + z^4 + z^2 + 1)(z^2 + z – z + 1)
= z^8 + z^6 + z^4 + z^4 + z^2 + z^2 + z^2 + z + z – z + 1
= z^8 + z^6 + 2z^4 + 3z^2 + 1
Comparing the simplified expressions, we find that option C: (z^4+1)(z^2+1)(z+1)(z–1) is the correct expression equivalent to z^8–1.
Therefore, the correct option is C.
To determine which expression is equivalent to z^8-1, we need to factor the given expression z^8-1 using the difference of squares formula.
The difference of squares formula states that a^2 - b^2 can be factored as (a + b)(a - b).
So, applying this formula to the given expression z^8-1, we have:
z^8 - 1 = (z^4)^2 - 1^2 = (z^4 + 1)(z^4 - 1).
Now we can further factor z^4 - 1 using the difference of squares again:
z^4 - 1 = (z^2)^2 - 1 = (z^2 + 1)(z^2 - 1).
Finally, we can factor z^2 - 1 using the difference of squares:
z^2 - 1 = (z + 1)(z - 1).
Hence, putting it all together, the factored form of z^8-1 is:
(z^4 + 1)(z^4 - 1) = (z^4 + 1)(z^2 + 1)(z^2 - 1) = (z^4 + 1)(z^2 + 1)(z + 1)(z - 1).
Comparing this factored form with the given options, we can see that option D, (z^4+1)(z^2+1)(z–1)(z–1), is the correct expression equivalent to z^8-1.
So, the correct answer is D: (z^4+1)(z^2+1)(z–1)(z–1).