Precalculus PreAP/D
Rev 2017
2.5A: Fundamental Theorem of Algebra
“I WILL …
Find all zeros of a polynomial equation.
Write a polynomial equation given some zeros.”
I. Theorems
A. Fundamental Theorem of Algebra is the number of solutions in a polynomial equation
with rational coefficients is equal to the degree of the polynomial when including all
complex solutions and solutions of multiplicity. That includes all real, imaginary, and
multiplicity roots.
B. Conjugate Root Theorem: All complex zeros (imaginaries and radical) come in conjugate
pairs. If a zero is given, use the conjugate and GO BACKWARDS.
C. Every _____________________ or _____________________ number has TWO roots
Ex 1: Write the simplest polynomial function
with the given zeros of 4, − 1, and −2
Ex 2: Write the simplest polynomial function
with the given zeros of 2, − 2, and 0.
Your Turn: Write the simplest polynomial
function with the given zeros of 3, 0, and −2
Ex 3: Determine the LINEAR FACTORS that
1
3
has 3, 2, and 2 are zeros. No fractions or
decimals are accepted.
Your Turn: Determine the LINEAR
1
1
FACTORS that has −2, 4, and – 2 are zeros.
No fractions or decimals are accepted.
Ex 4: Find a second degree polynomial
function with real coefficients that has √2 as a
zero.
Precalculus PreAP/D
Rev 2017
Ex 5: Find a fourth degree polynomial
function with real coefficients that has −
1 𝐷𝑅, and 3𝑖 are zeros.
Your Turn: Find a fifth degree polynomial
function with the given zeros of ±1, −2 and
−2𝑖
Ex 6: Write the simplest function with zeros
of 2 + 𝑖 and √3
Ex 7: Write the simplest function with zeros
of 2 and 3 + 𝑖
Ex 8: Write the simplest function with zeros
of 1 − 3𝑖, 3, and −2
Your Turn: Write the simplest function with
zeros of 5 and 3 − 2𝑖
Precalculus PreAP/D
Rev 2017
II. Finding Zeros
A. Use the rational root theorem (𝑃(𝑥) over 𝑄(𝑥)) to make a list of potential answers.
B. DIVIDE the function using
C. Do SYNTHETIC DIVISION or LONG DIVISION using these zeros to until you got it
down to a QUADRATIC equation.
Ex 9: Find all the zeros of 𝑓(𝑥) = 𝑥 4 −
Ex 10: Find all the zeros of 𝑓(𝑥) = 𝑥 4 −
3𝑥 3 + 6𝑥 2 + 2𝑥 − 60 given that 1 + 3𝑖 is a
5𝑥 3 + 4𝑥 2 + 2𝑥 − 8 given that 1 + 𝑖 is a
zero of 𝑓.
zero of 𝑓.
Ex 11: Find all the zeros of 𝑓(𝑥) = 𝑥 3 −
4𝑥 2 + 21𝑥 − 34 given that 1 + 4𝑖 is a zero
of 𝑓.
Assignment: Worksheet
Your Turn: Find all the zeros of 𝑓(𝑥) = 𝑥 3 −
11𝑥 2 + 41𝑥 − 51 given that 4 − 𝑖 is a zero
of 𝑓.