solve x^5-16=0 completely given that 2 solutions is 1 (hint there are 4 zeros)
the real solution is x = cuberoot(16)
there are 4 complex roots, two of which are
(-2^.8 - 8000^.1)/4 ± 2sin(pi/5)i
check your typing, I don't really understand what you mean by
"given that 2 solutions is 1 "
instead of <<the real solution is x = cuberoot(16) >>
I meant to say
the real solution is x = 5throot(16)
To solve the equation x^5 - 16 = 0, we can first group the equation as the difference of two squares:
(x^5)^2 - 4^2 = 0
Now, we can apply the difference of squares formula, which states that a^2 - b^2 can be factored as (a + b)(a - b):
(x^5 + 4)(x^5 - 4) = 0
Now, we have factored the equation into two parts. To solve for x, we set each part equal to zero and solve for x separately:
x^5 + 4 = 0
To solve this equation, we subtract 4 from both sides:
x^5 = -4
To find the fifth root of -4, we can rewrite the equation as follows:
x = (-4)^(1/5) (1/5 represents the fifth root)
The fifth root of a negative number can be expressed using complex numbers. In this case, the fifth root of -4 is:
x = (-4)^(1/5) = 1 + i√3 (where i represents the imaginary unit)
Now, let's solve the second part:
x^5 - 4 = 0
To solve this, we add 4 to both sides:
x^5 = 4
Taking the fifth root of 4 gives us the values of x:
x = 4^(1/5)
Since the question states that two solutions are 1, we can rewrite the equation as follows:
(x - 1)(x - 1)(x - (4^(1/5)))(x + (4^(1/5)))(x - (1 + i√3))(x - (1 - i√3)) = 0
Thus, the complete solution to x^5 - 16 = 0 is:
x = 1, 1, 4^(1/5), -4^(1/5), 1 + i√3, 1 - i√3