Question

Create a polynomial of degree 6 that has no real roots. Explain why it has no real roots. Is it possible to have a polynomial with an odd degree that has no real roots? Explain.

Answer 1

Create a polynomial of degree 6 that has no real roots.

y = ( x^2 + 4) ( x^2 +7 ) ( x^2+5)

Multiplying all the terms together

y =x ^6 + 16 x^4 + 83 x^2 + 140

Using the zero product property

0= x^2 +4 x^2+7 =0 x^2 + 5 =0 will each give a complex solution

x^2 = -4 x^2 = -7 x^2 = -5

This means x = 2i or -2i x = i sqrt(7) or -i sqrt(7) x = i sqrt (5) or - i sqrt(5)

These solutions can be in the form a+bi

Therefor it will have no real roots

y = x^6 + 16 x^4 + 83 x^2 + 140 has no real solutions

Complex solutions come in pairs, so an odd degree must have a real solution