Question 720586: Give an example of a polynomial function that has no real zeros. Explain how you came up with your function. Answer by jsmallt9(3758) (Show Source):
If there are no real zeros, then the zeros must be complex numbers (of the form a + bi).
If a polynomial with real coefficients has complex roots, then those roots will come in complex conjugate pairs. The complex conjugate of a + bi is a + (-bi) (or a - bi). (You don't mention that the polynomial should have real coefficients but I think it is a safe assumption.)
If "z" is a zero of a polynomial then (x-z) will be a factor of that polynomial.
So to find a polynomial with no real roots:
Pick a complex number to be a zero of the polynomial. (Hint: Pick a complex number whose "a" is zero. For example: 0 + 4i (which is just 4i))
Find the complex conjugate of the number you picked in step 1. This will be another zero of the polynomial.
Repeat steps 1 and 2 as many times as you want (including zero times). Each repetition will get you two more zeros of the polynomial. (If you're lazy then don't repeat and just use two zeros. If you want to impress, find two or three pairs of zeros.)
Write the polynomial function in factored form. Make a factor of the form (x - z) for each of the zeros you have. So the polynomial function will look like:
P(x) = (x-z)*...
with as many factors as you have zeros. Since the zeros come in conjugate pairs there will be an even number of these factors.
Multiply all the factors. If you have chosen complex zeros with a's that are zero (like I suggested), then you will have pairs of factors that look like:
(x-z)(x+z)
These pairs can be multiplied quickly and easily by using the
pattern. Whether you use the pattern or use FOIL you will end up with some 's. Remember that . So you will replace the 's with -1's. When you are finished multiplying all your factors you will have your answer.
P.S. If you choose complex roots with a's that are not zeros...
Let's use an example to illustrate how this would work. Let's choose 2 + 3i for our zero. Then 2 - 3i will also be a zero. This makes the factors for these zeros:
(x - (2 + 3i)) and (x - (2 - 3i))
These simplify to :
(x - 2 - 3i)) and (x - 2 + 3i))
Believe it or not, these factors can still be multiplied using the
pattern. You just have to treat "x-2" as the "a" of the pattern and the "3i" as the "b". So according to the pattern
is equal to:
We can use another pattern, , on :
(