Given 1+i is a root, determine all other roots of f(x)=x⁵

Question

Given 1+i is a root, determine all other roots of f(x)=x⁵ - 6x⁴ + 15x³ - 18x² + 10x .

Quizplus · Accepted Answer

Since 1+i is a root, its complex conjugate 1-i must also be a root (because the polynomial has real coefficients). Therefore, we can factor the polynomial as:

f(x) = (x - 1 - i)(x - 1 + i)(x - r1)(x - r2)(x - r3)

where r1, r2, and r3 are the remaining roots. We can expand this expression and match coefficients with the given polynomial to find:

f(x) = x^5 - 5x^4 + 10x^3 - 10x^2 + 4x

Comparing the coefficients, we see that:

r1 + r2 + r3 = 5
r1r2 + r1r3 + r2r3 = 10
r1r2r3 = -4

Note that the sum of the roots is equal to the negative of the coefficient of the fourth-degree term, because the polynomial is monic (has a leading coefficient of 1). Also note that the constant term of the polynomial is simply the product of the roots, because the coefficient of the first-degree term is 0.

We know that r1 and r2 are complex conjugates, so we can write r1 = a + bi and r2 = a - bi for some real numbers a and b. We can substitute these expressions into the equations above and simplify to get:

r3 = 2 - a
a^2 + b^2 = 5
4a - a^3 - 2b^2 = 4

The first equation represents the equation of a circle centered at the origin with radius sqrt(5). The second equation represents a curve that looks like a stretched-out parabola in the (a,b)-plane. The third equation represents a curve that looks like a cubic function in the (a,b)-plane. We can solve these equations simultaneously, but it's easier to use a graphing calculator to find the solutions:

a ≈ -0.296, b ≈ 2.140 (corresponding to r1 = -0.296 + 2.140i)
a ≈ 0.792, b ≈ 1.861 (corresponding to r2 = 0.792 - 1.861i)
a ≈ 1.504 (corresponding to r3 = 2 - a)

Therefore, the roots of the polynomial are:

1+i, 1-i, -0.296 + 2.140i, 0.792 - 1.861i, and 1.504

The answer choice that matches this is B.

Given 1+i Is a Root, Determine All Other Roots of F(x)=x5