Download Precalculus PreAP/D Rev 2017 2.5: Rational Zero Test “I WILL

Precalculus PreAP/D
Rev 2017
2.5: Rational Zero Test
“I WILL…
…find all rational zeros of a polynomial function.
…apply the Rational Root Theorem.”
I. Finding Rational Zeros
A. Where the graphs cross the 𝑥–axis (𝑦 is zero)
B. Known as zeros or roots
C. To find the real–number zeros, it can be rational, irrational and/or imaginary numbers
D. To find out how many zeros the graph has, the easiest way to find the amount of degrees
the graphs has
II. Rational Root Theorem
A. Identify the constant term and coefficient of the polynomial
B. List all multiples of each number and don’t forget the plus-or-minus symbol for each
polynomial
C. Apply the equation: =
D. List ALL of the possible rational zeros
Ex 1: List all possible rational zeros for 𝑥 8 +
18𝑥 4 − 9𝑥 2 + 2 = 0
Ex 2: List all possible rational zeros for
𝑓(𝑥) = 2𝑥 4 + 𝑥 3 − 17𝑥 2 − 4𝑥 + 6
Your Turn: Find all possible rational zeros for 4𝑥 4 − 21𝑥 3 + 18𝑥 2 + 19𝑥 − 6 = 0
III. Steps in finding rational zeros
A. Determine what the leading coefficient and constant term are.
B. Put it in the rational zero test (p/q)
C. Use synthetic division to find which factor gives the remainder of zero.
1. If the polynomial cannot be factored, repeat steps 2 and 3 again
D. Factor or use the Quadratic Formula to get the leftover polynomial out.
E. Equal it all to zero and put it in x= form.
Precalculus PreAP/D
Rev 2017
Ex 3: Find the real and imaginary rational
zeros for 𝑓(𝑥) = 𝑥 3 + 𝑥 2 − 2𝑥 − 2
Ex 4: Find the real and imaginary rational
zeros for 𝑓(𝑥) = 3𝑥 3 + 7𝑥 2 − 22𝑥 − 8
Ex 5: Find the real and imaginary rational
zeros for 𝑓(𝑥) = 𝑥 3 − 7𝑥 2 + 11𝑥 − 5
Your Turn: Find the real and imaginary
rational zeros for 𝑓(𝑥) = 6𝑥 3 + 𝑥 2 − 10𝑥 +
3
Precalculus PreAP/D
Rev 2017
Ex 6: Find the real and imaginary rational
zeros for 𝑓(𝑥) = 2𝑥 4 + 𝑥 3 − 17𝑥 2 − 4𝑥 + 6
Ex 7: Find the real and imaginary rational
zeros for 𝑓(𝑥) = 𝑥 4 − 𝑥 3 − 𝑥 2 − 𝑥 − 2
Your Turn: Find the real and imaginary rational zeros for 𝑓(𝑥) = 𝑥 4 − 𝑥 3 − 𝑥 2 − 𝑥 − 2
Assignment: Worksheet