Find all solutions of:
z^3-8p^3=0
Where p is a positive real number
Write your answers in polar form as expressions in terms of p
z^3 = 8p^3 cis2π
z = 2p cis 2π/3, 2p cis 4π/3, 2p cis 0
To find all solutions of the equation z^3 - 8p^3 = 0, we can rewrite it as:
z^3 = 8p^3
Now, let's express the complex number z in polar form, z = r(cosθ + isinθ), where r is the modulus (or magnitude) of z, and θ is the argument (or angle).
Substituting this into the equation, we get:
[r(cosθ + isinθ)]^3 = 8p^3
Expanding the cube, we have:
r^3(cos^3θ + 3cos^2θisinθ - 3cosθsin^2θ - isin^3θ) = 8p^3
Since r is a positive real number, we can rewrite r^3 as (r^3)^(1/3) = r, and simplify the equation further:
r^3(cos^3θ - 3cosθsin^2θ + 3cos^2θisinθ - isin^3θ) = 8p^3
Now, we equate the real and imaginary parts separately:
Real part: r^3(cos^3θ - 3cosθsin^2θ) = 8p^3
Imaginary part: r^3(3cos^2θisinθ - isin^3θ) = 0
Let's solve the imaginary part first:
r^3(3cos^2θisinθ - isin^3θ) = 0
Since r is a positive real number, this equation implies that either the real part or the imaginary part must be zero. Therefore, we have two cases to consider:
Case 1: 3cos^2θisinθ = 0
This equation is satisfied when either cos^2θ = 0 or sinθ = 0. If cos^2θ = 0, then cosθ = 0, which implies that θ = π/2 or θ = 3π/2. If sinθ = 0, then θ = 0 or θ = π. Thus, the solutions in this case are:
z = r(cos(π/2) + isin(π/2)) = r(i) or z = r(cos(3π/2) + isin(3π/2)) = -r(i)
Case 2: -isin^3θ = 0
This equation is satisfied when sin^3θ = 0. If sin^3θ = 0, then sinθ = 0, which implies that θ = 0 or θ = π. Thus, the solutions in this case are:
z = r(cos(0) + isin(0)) = r or z = r(cos(π) + isin(π)) = -r
Now, let's solve the real part equation:
r^3(cos^3θ - 3cosθsin^2θ) = 8p^3
Simplifying this equation, we get:
cos^3θ - 3cosθsin^2θ = 8p^3/r^3
Since θ can take any angle in polar form, it is not possible to generalize the solutions explicitly in terms of p alone. The solutions will depend on the specific value of p.
Thus, the solutions of the equation z^3 - 8p^3 = 0, in polar form as expressions in terms of p, are:
z = r, -r, ri, or -ri, where r is a positive real number.