http://www.ck12.org Chapter 11. Complex Numbers
- Keep in mind that all polynomials can be written in factorized form like the above polynomial, due to a theorem
called the Linear Factorization Theorem.
Example A
Identify the zeroes of the following complex polynomial.
f(x) =x^2 + 9
Solution:Sety=0 and solve forx. This will give you the two zeros.
0 =x^2 + 9
− 9 =x^2
± 3 i=xThus the linear factorization of the function is:
f(x) = (x− 3 i)(x+ 3 i)
Example B
Examine the following graph and make conclusions about the number and type of zeros of this 7thdegree polynomial.
Solution:A 7thdegree polynomial has 7 roots. Three real roots are visible. The root atx=−2 has multiplicity 2
and the root atx=2 has multiplicity 1. The other four roots appear to be imaginary and the clues are the relative
maximums and minimums that do not cross thexaxis.
Example C
Identify the polynomial that has the following five roots.x= 0 , 2 , 3 ,±
√
5 i
Solution:Write the function in factorized form.
f(x) = (x− 0 )(x− 2 )(x− 3 )(x−
√
5 i)(x+√
5 i)
When you multiply through, it will be helpful to do the complex conjugates first. The complex conjugates are√ (x−
5 i)(x+√ 5 i).
f(x) =x(x^2 − 5 x+ 6 )(x^2 − 5 ·(− 1 ))
f(x) = (x^3 − 5 x^2 + 6 x)(x^2 + 5 )
f(x) =x^5 − 5 x^4 + 6 x^3 + 5 x^3 − 25 x^2 + 30 x
f(x) =x^5 − 5 x^4 + 11 x^3 − 25 x^2 + 30 x