11.1. Fundamental Theorem of Algebra http://www.ck12.org
Concept Problem Revisited
When a parabola fails to cross thexaxis it still has 2 roots. These two roots happen to be imaginary numbers. The
functionf(x) =x^2 +4 does not cross thexaxis, but its roots arex=± 2 i.
Vocabulary
Acomplex numberis a number written in the forma+biwhere bothaandbare real numbers. Whenb=0, the
result is a real number and whena=0 the result is an imaginary number.
Animaginary numberis the square root of a negative number. √−1 is defined to be the imaginary numberi.
Complex conjugatesare pairs of complex numbers with real parts that are identical and imaginary parts that are of
equal magnitude but opposite signs. 1+ 3 iand 1− 3 ior 5iand− 5 iare examples of complex conjugates.
Guided Practice
- Write the polynomial that has the following roots: 4 (with multiplicity 3), 2 (with multiplicity 2) and 0.
- Factor the following polynomial into its linear factorization and state all of its roots.
f(x) =x^4 − 5 x^3 + 7 x^2 − 5 x+ 6 - Can you create a polynomial with real coefficients that has one imaginary root? Why or why not?
Answers:
1.f(x) = (x− 4 )^3 ·(x− 2 )^2 ·x - You can use polynomial long division to obtain the following factorization.
f(x) = (x− 3 )(x− 2 )(x−i)(x+i)
If you need a place to start, it is helpful to look at the graph of the polynomial and notice that the graph shows you
exactly where the real roots appear. - No, if a polynomial has real coefficients then either it has no imaginary roots, or the imaginary roots come in pairs
of complex conjugates (so that the imaginary portions cancel out when the factors are multiplied).
Practice
For 1 - 4, find the polynomial with the given roots.
- 2 (with multiplicity 2), 4 (with multiplicity 3), 1,
√
2 i,−√
2 i.