Given that −8i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. f(x)=x4+10x3+85x2+640x+1344

Answer 1

The Conjugate Roots Theorem states that imaginary roots come in pairs. If -8i is a root of the polynomial, then so is +8i.

If both -8i and +8i are factors, we rewrite them as (x + 8i) and (x - 8i) in factored form. Since imaginary terms aren't accepted, you can FOIL these to get a quadratic with no imaginary terms.

x² - 8ix + 8ix - 64i²

Simplify. Replace i² with -1 because √-1 = i

x² - 64(-1)

x² + 64

The quadratic x² + 64 has roots -8i and 8i. You can use the quadratic formula to verify this.

Now, let's tackle the rest of the polynomial. If x² + 64 is a factor of the polynomial

x⁴ + 10x³ + 85x² + 640x + 1344, divide it out to see what is left. I will use long division of polynomials. Note, I am using the square root symbol as a division symbol.

x² + 10x + 21

x² + 64 √ (x⁴ + 10x³ + 85x² + 640x + 1344)

x⁴ + 64x²

_____________

10x³ + 21x²

10x³ + 640x

__________

21x² +1344

21x² + 1344

___________

0

There is no remainder, meaning x² + 64 went in evenly. The quotient left when we divide out x² + 64 from

x⁴ + 10x³ + 85x² + 640x + 1344 is x² + 10x + 21. Factor the quotient to get (x + 7)(x + 3).

x⁴ + 10x³ + 85x² + 640x + 1344 = (x² + 64)(x + 7)(x + 3)

You can check this factored form by FOILing out, and you should get the original polynomial.