4x4 – 13x3 + 8x2 – 21x + 18 is divided by 2 – 3?

x = 3/4 or x = ((-1)^(1/3) (127 - 18 sqrt(43))^(2/3) - 13 (-1)^(2/3))/(3 (127 - 18 sqrt(43))^(1/3)) - 4/3 or x = 1/3 (13 (-1/(127 - 18 sqrt(43)))^(1/3) - (-1)^(1/3) (18 sqrt(43) - 127)^(1/3)) - 4/3 or x = (-(127 - 18 sqrt(43))^(2/3) - 13)/(3 (127 - 18 sqrt(43))^(1/3)) - 4/3

Explanation:

Solve for x over the real numbers:

4 x^4 + 13 x^3 - 8 x^2 + 21 x - 18 = 0

The left hand side factors into a product with two terms:

(4 x - 3) (x^3 + 4 x^2 + x + 6) = 0

Split into two equations:

4 x - 3 = 0 or x^3 + 4 x^2 + x + 6 = 0

Add 3 to both sides:

4 x = 3 or x^3 + 4 x^2 + x + 6 = 0

Divide both sides by 4:

x = 3/4 or x^3 + 4 x^2 + x + 6 = 0

Eliminate the quadratic term by substituting y = x + 4/3:

x = 3/4 or 14/3 + 4 (y - 4/3)^2 + (y - 4/3)^3 + y = 0

Expand out terms of the left hand side:

x = 3/4 or y^3 - (13 y)/3 + 254/27 = 0

Change coordinates by substituting y = z + λ/z, where λ is a constant value that will be determined later:

x = 3/4 or 254/27 - 13/3 (z + λ/z) + (z + λ/z)^3 = 0

Multiply both sides by z^3 and collect in terms of z:

x = 3/4 or z^6 + z^4 (3 λ - 13/3) + (254 z^3)/27 + z^2 (3 λ^2 - (13 λ)/3) + λ^3 = 0

Substitute λ = 13/9 and then u = z^3, yielding a quadratic equation in the variable u:

x = 3/4 or u^2 + (254 u)/27 + 2197/729 = 0

Find the positive solution to the quadratic equation:

x = 3/4 or u = 1/27 (18 sqrt(43) - 127)

Substitute back for u = z^3:

x = 3/4 or z^3 = 1/27 (18 sqrt(43) - 127)

Taking cube roots gives 1/3 (18 sqrt(43) - 127)^(1/3) times the third roots of unity:

x = 3/4 or z = 1/3 (18 sqrt(43) - 127)^(1/3) or z = -1/3 (-1)^(1/3) (18 sqrt(43) - 127)^(1/3) or z = 1/3 (-1)^(2/3) (18 sqrt(43) - 127)^(1/3)

Substitute each value of z into y = z + 13/(9 z):

x = 3/4 or y = 1/3 (18 sqrt(43) - 127)^(1/3) - (13 (-1)^(2/3))/(3 (127 - 18 sqrt(43))^(1/3)) or y = 13/3 ((-1)/(127 - 18 sqrt(43)))^(1/3) - 1/3 (-1)^(1/3) (18 sqrt(43) - 127)^(1/3) or y = 1/3 (-1)^(2/3) (18 sqrt(43) - 127)^(1/3) - 13/(3 (127 - 18 sqrt(43))^(1/3))

Bring each solution to a common denominator and simplify:

x = 3/4 or y = ((-1)^(1/3) (127 - 18 sqrt(43))^(2/3) - 13 (-1)^(2/3))/(3 (127 - 18 sqrt(43))^(1/3)) or y = 1/3 (13 ((-1)/(127 - 18 sqrt(43)))^(1/3) - (-1)^(1/3) (18 sqrt(43) - 127)^(1/3)) or y = (-(127 - 18 sqrt(43))^(2/3) - 13)/(3 (127 - 18 sqrt(43))^(1/3))

Substitute back for x = y - 4/3:

x = 3/4 or x = ((-1)^(1/3) (127 - 18 sqrt(43))^(2/3) - 13 (-1)^(2/3))/(3 (127 - 18 sqrt(43))^(1/3)) - 4/3 or x = 1/3 (13 (-1/(127 - 18 sqrt(43)))^(1/3) - (-1)^(1/3) (18 sqrt(43) - 127)^(1/3)) - 4/3 or x = (-(127 - 18 sqrt(43))^(2/3) - 13)/(3 (127 - 18 sqrt(43))^(1/3)) - 4/3

c)as speed changes, wavelength and frequency change.

everything changes!

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