3.4: Zeros of Polynomial Functions
Warm-up
1
Solve the equation 3𝑥 3 + 7𝑥 2 − 22𝑥 − 8 = 0 given that − is
3
a root.
Today we will be finding a lot of zeros. You may ask yourself:
There exists a way to quickly determine all possible rational zeros of
any function:
In other words:
1. Find all the possible rational zeros of:
a) 𝑓(𝑥 ) = 𝑥 3 + 3𝑥 2 − 6𝑥 − 8
b) ℎ(𝑥 ) = 2𝑥 4 + 3𝑥 3 − 11𝑥 2 − 9𝑥 + 15
c) 𝑣(𝑥 ) = 4𝑥 5 − 8𝑥 4 − 𝑥 − 2
Finding the zeros of a polynomial function
Step 1: Find all possible rational zeros with the Rational Zero
Theorem
Step 2: Check possible zeros synthetic division or with the
Factor Theorem: if 𝑓 (𝑐) = 0, then 𝑥 − 𝑐 is a factor of 𝑓(𝑥)
Step 3: Use synthetic division to factor the polynomial
Step 4: Continue to factor the remaining polynomial.
2. Find all zeros of the polynomial functions.
a. 𝑓(𝑥 ) = 𝑥 3 − 2𝑥 2 − 11𝑥 + 12
b. 𝑓(𝑥 ) = 2𝑥 3 − 5𝑥 2 + 𝑥 + 2
c. 𝑓(𝑥 ) = 2𝑥 3 + 𝑥 2 − 3𝑥 + 1
We can also use this same approach to solve polynomial
equations.
3. Find all roots of the polynomial equation.
a. 𝑥 3 − 2𝑥 2 − 7𝑥 − 4 = 0
b. 2𝑥 3 − 5𝑥 2 − 6𝑥 + 4 = 0
c. 𝑥 4 − 2𝑥 2 − 16𝑥 − 15 = 0
4. Find an 3𝑟𝑑 degree polynomial with zeros of 4 𝑎𝑛𝑑 2𝑖
such that 𝑓 (−1) = −50
5. Find an 3𝑟𝑑 degree polynomial with zeros of
−4 and 2 + 3i such that 𝑓(2) = 9.
Okay, let’s put it all together.
6. Find all complex zeros of the polynomial function.
a. 𝑓(𝑥 ) = 2𝑥 4 + 3𝑥 3 − 11𝑥 2 − 9𝑥 + 15
b. 𝑓(𝑥 ) = 3𝑥 4 − 11𝑥 3 − 3𝑥 2 − 6𝑥 + 8
c. 𝑔(𝑥 ) = 4𝑥 5 + 12𝑥 4 − 41𝑥 3 − 99𝑥 2 + 10𝑥 + 24