Question

The polynomial equation x5-16x2=4x4-64 has complex roots + or - 2i what are the other roots? use a graphing calculator and a system of equations
A. -64, 0
B. -2, 2
C. -1, 1
D. 0, 64

Answer 1

Kindly proofread the question!!!

The leading term is x^6, not x5.

Answer: The additional roots are -2, +2, each with multiplicity 2.

Step-by-step explanation:

x^6 - 16x^2 = 4x^4 - 64

x^6 - 4x^4 - 16x^2 + 64 = 0

(x+2i)(x-2i) = (x^2+4) is a factor.

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

Polynomial long division by x^2+4

x^6 - 4x^4 - 16x^2 + 64 = 0

First term of quotient is x^4

Subtract x^6+4x^4

Remainder is -8x^4 -16x^2+64

Second quotient term is -8x^2

Subtract -8x^4-32x^2

Remainder is 16x^2+64

Third quotient term is 16

Subtract 16x^2+64

Remainder is zero

Quotient is x^4 - 8x^2 + 16

(x^2-4)^2 = x^4 + 2(1)(-4)x^2 + 16

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