Warm Up 9/2
• What is the remainder of
2𝑥 2 −3𝑥+1
𝑥−2
?
• Is x-1 a factor of the polynomial 𝑥 3 − 𝑥 2 + 𝑥 − 1 ?
Homework Questions
Announcements
• Quiz Friday
• HW 2.4 pg. 206 #’s 34-44 even, 50-56 even
Rational Zeros Theorem
• To find all possible rational roots, divide all the factors of the constant
by all the factors of the leading coefficient.
Example
• Find the rational zeros for f x = 6𝑥 3 − 5𝑥 − 1
Practice
• Find the rational zeros for f(x)=3𝑥 3 + 4𝑥 2 − 5𝑥 − 2
Upper and Lower Bounds
• Upper Bound – If every number in last line of synthetic division is positive
• Lower Bound – If the numbers in last line of synthetic division alternate
between positive and negative.
Example
• Prove that all the real zeros of f(x) = 2𝑥 4 − 7𝑥 3 − 8𝑥 2 + 14𝑥 + 8 must lie
in the interval [-2, 5]
Practice
• Prove that 3 is an upper bound for 𝑓 𝑥 = 2𝑥 3 − 4𝑥 2 + 𝑥 − 2
• Prove that -1 is a lower bound for 𝑓 𝑥 = 3𝑥 3 − 4𝑥 2 + 𝑥 + 3
Irrational Roots
• Find all real zeros of 𝑓 𝑥 = 𝑥 3 − 6𝑥 2 + 7𝑥 + 4 and tell which are rational
and irrational
Practice
• Find all real zeros of 𝑓 𝑥 = 2𝑥 3 − 3𝑥 2 − 4𝑥 + 6 and tell which are
rational and irrational
Homework
• Section 2.4 pg. 206 #’s 34-44 even, 50-56 even
• Answers posted on website