College Algebra
Chapter 4, section 5
Created by Lauren Atkinson
Mary Stangler Center for Academic Success
This review is meant to highlight basic concepts from Chapter 4. It does not cover all concepts presented by your instructor. Refer back
to your notes, handouts, the book, MyMathLab, etc. for further prepare for your exam.
4.5: The Fundamental Theorem of Algebra
• This section covers:
– The Fundamental Theorem of Algebra
– The Number of Zeros Theorem
– Complex zeros
The Fundamental Theorem of Algebra
• A polynomial function with degree of 1 or
larger, has at least one complex zero.
Degree of polynomials=
number of real zeros + number of complex zeros
• Real Zero= graph either hits or crosses through
the x-axis
• Complex zero= graph doesn’t hit or cross
through the x-axis
f ( x)  ( x  2)  3
2
All of these functions
have a degree of 2,
therefore they, at most,
have 2 zeros.
This graph doesn’t touch the
x-axis or cross it at all,
therefore it has 2 imaginary
zeros (0 real zeros)
f ( x)  ( x  2) 2
This graph bounces off the xaxis, therefore it has 1 real
zero and one imaginary zero
f ( x)  ( x  2) 2  4
This graph touches the x-axis
twice, therefore it has 2 real
zeros (0 imaginary zeros)
Number of Zeros Theorem
• A polynomial of degree n has at most n
distinct zeros (x-intercepts, factors)
• Examples:
Find the complete factored form of a polynomial with real coefficients 𝑓(𝑥) that
satisfies the conditions:
Degree 4; 𝑎𝑛 = 10; zeros 1,-1,3𝑖, −3𝑖
Degree 2; 𝑎𝑛 = -5; zeros 1 + 𝑖, 1 − 𝑖
= 10(𝑥 − 1)(𝑥 + 1)(𝑥 − 3𝑖)(𝑥 + 3𝑖)
= −5 𝑥 − 1 + 𝑖
𝑥− 1−𝑖
= −5(𝑥 − 1 − 𝑖)(𝑥 − 1 + 𝑖)
Complex Zeros:
• It is true that if 𝑎 + 𝑏𝑖 is a zero, then the
conjugate (𝑎 − 𝑏𝑖) is also a zero.
• Examples:
If one zero is −2𝑖 then another zero (by default) is 2𝑖.
Similarly, if one zero is 1 + 3𝑖 then another zero is
− 1 + 3𝑖 = −1 − 3𝑖
Additional Examples:
Find all zeros of 𝑓 𝑥 :
𝑓 𝑥 = 3𝑥 3 + 3𝑥
= 3𝑥 𝑥 2 + 1 = 3𝑥 𝑥 + 𝑖 𝑥 − 𝑖
Zeros=0, ±𝑖
𝑓 𝑥 = 𝑥 3 + 2𝑥 2 + 16𝑥 + 32
= 𝑥 2 𝑥 + 2 + 16 𝑥 + 2 =
𝑥 + 2 𝑥 2 + 16 = (𝑥 + 2)(𝑥 + 4𝑖)(𝑥 − 4𝑖)
Zeros= −2, ±4𝑖
The whole point …
• The sections we have covered so far are
supposed to prepare you to solve all
equations, which is why a majority of the
problems in section 4.5 ask you to solve
polynomial equations.
Examples:
Solve:
𝑥 3 = 2𝑥 2 − 7𝑥 + 14
𝑥 3 − 2𝑥 2 + 7𝑥 − 14 = 0
𝑥2 𝑥 − 2 + 7 𝑥 − 2 = 0
𝑥 − 2 𝑥2 + 7 = 0
𝑥 = 2, ±𝑖 7
Solve:
𝑥 4 − 2𝑥 3 + 𝑥 2 − 2𝑥 = 0
𝑥 3 𝑥 − 2 + 𝑥(𝑥 − 2) = 0
(𝑥 3 + 𝑥) 𝑥 − 2 = 0
𝑥 𝑥2 + 1 𝑥 − 2 = 0
𝑥 𝑥+𝑖 𝑥−𝑖 𝑥−2 =0
𝑥 = 0, ±𝑖, 2