Finding Roots of Polynomials
Warm Up- CALCULATOR INACTIVE!
Sketch the following….
A
A cubic function with a
negative zero of multiplicity 2
and a positive zero
C
A cubic function with no real
zeroes
B
A quartic function with a
negative leading coefficient, a
positive y intercept, one
negative double root, one
positive zero, and one zero at
the origin.
D
A quartic function with no real
zeroes, a positive leading
coefficient, and a positive y
intercept
Finding From an Equation
Factor Theorem:
𝑓(𝑐) = 0 iff 𝑥 − 𝑐 is a factor of the polynomials
Thus, c is a root of the polynomial
Find all roots of the following polynomials:
1) 𝑓 𝑥 = (𝑥 − 2)(𝑥 + 1)
3) 𝑓 𝑥 = 𝑥 ! + 19𝑥 + 34
2) 𝑓 𝑥 = 𝑥 − 3 ! (2𝑥 − 1)(𝑥 + 2)
4) 𝑓 𝑥 = 𝑥 ! + 2𝑥 ! + 𝑥
“Fancy Factoring”
By examining the form of the following polynomial, solve by factoring:
𝑥 ! + 7𝑥 ! = 19
1. 𝑥 ! − 10𝑥 ! = −9
2. 𝑥 ! − 8𝑥 ! = −16
3. 𝑥 ! − 12𝑥 ! = 64
4. 𝑥 ! − 16 = 0
Write a polynomial with given zeroes:
5, 3i
Finding Roots using Polynomial Division
Function
Rational
Roots
𝟑
𝟐
𝒇 𝒙 = 𝒙 − 𝟓𝒙 + 𝟓𝒙 − 𝟒
− 3, 2
Find Other Roots
𝑓 𝑥 = 𝑥 ! + 2𝑥 ! − 4𝑥 ! − 7𝑥 − 2
𝑓 𝑥 = 𝑥 ! − 27
Pairs Task:
Given the polynomial y = -2(x + 1)2(x – 2)(x – 3)3(x2 + 2) determine the following without graphing.
a) Determine the quadrants where the graph originates/terminates.
b) Determine the zeros of the function.
c) Determine the x-intercepts of the function.
d) Determine the y-intercept of the function.
e) Describe the behavior of the graph at each of the x-intercepts.
f) Without using the technology, sketch the graph of the function
Homework!
1. 𝑓 𝑥 = 𝑥 ! + 2𝑥 ! − 5𝑥 − 10
2. 𝑓 𝑥 = 𝑥 ! + 3𝑥 ! + 𝑥 ! − 12𝑥 − 20
3. 𝑓 𝑥 = 2𝑥 ! + 3𝑥 ! + 18𝑥 + 27
4. 𝑓 𝑥 = 𝑥 ! + 64
5. 𝑓 𝑥 = 𝑥 ! + 4𝑥 ! + 7𝑥 + 28
6. 𝑓 𝑥 = 2𝑥 ! + 𝑥 ! + 1