Question

PLEASE HELP:
Write a polynomial of degree 3 that has no real zeros.

(Give an example of a polynomial that has the given properties, or explain why it is impossible to find such a polynomial)

Answer 1

A polynomial of degree 3, or any odd degree, must have a real root. If you think about the value going from minus infinity to plus infinity it will cross the x axis and if you switch x with minus x the reverse will be true so an odd degree polynomial always crosses the x-axis

Answer 2

so this is impossible?

Answer 3

Correct. It is not possible to have an odd degree polynomial with no real roots

Answer 4

think about it analytically

\[x^2+1\] if you try and find a root of this you cannot but \[x^3+1\] you can find a real root of.

Answer 5

Ok thanks. Do you think you could help me with another question like that?

Find a polynomial of degree 4 that has no real zeros

Answer 6

Try starting by finding polynomials of degree 2 which have no real roots and see what you can do with those. Try that and ask if you need further help

Answer 7

like (x^2+1) and (x^2+4) ?

Answer 8

For example yes. Now try things like multiplying them together, or multiplying one of those with a polynomial that does have real roots and see what results. Mathematics requires exploration and experimentation

Answer 9

so if i did (x^2+1)(x^2+1) which expands to x^4+2x^2+1 then the roots would not be real right? the factors would be (x-i)^2 and (x+i)^2 ?

Answer 10

Make sure to check all of your multiplication and check that there are no real roots. I wasn't tell you that two polynomials of degree 2 multiplied together give a polynomial of degree 4 with no real roots I was just suggesting an avenue for you to explore. Multiply things out and then factor them and see if you can find a real root.