Precalculus Exam 1 Review (Unit 3): Polynomial Functions and Rational Functions

#precalculus #polynomials #rationalfunctions

Links and resources
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(0:00) Graph y=(x-1)^3+8
(4:32) Graph y=-x^3+x^2+12x
(8:41) Find P(3) using Remainder Theorem
(10:56) Polynomial long division to find answer
(13:39) Synthetic division to find answer
(15:39) Why does this work?
(17:07) Graph a quartic polynomial, find complete factorization, and describe end behavior.
(19:01) Rational Root Theorem
(26:13) The graph
(28:30) Descartes' Rule of Signs for the number of positive real roots of a polynomial
(31:05) Show 9 is an upper bound for the real roots and is itself not a root
(33:17) Show 2 is a double root (root of multiplicity two)
(34:47) Graph P(x) and find local extrema with technology
(39:31) Find quotient and remainder for a rational function
(41:52) Graph the quotient
(42:39) Graph the rational function
(44:40) Graph a rational function by factoring without technology and describe its end behavior. It has a removable discontinuity where the function is undefined but the limit exists.
(50:22) Removable discontinuity and limiting value
(51:57) Evaluate i^15 (i is the imaginary unit: i^2 = -1)
(54:51) Complex division: Write (4-3i)/(5-2i) in the form a + bi. Use complex conjugate.
(57:52) Expand (2 - i)^4 into the form a + bi. Use Pascal's Triangle (Binomial Theorem).
(1:01:10) Find rational, irrational, and complex roots of a quartic polynomial. Then completely factor it. Discuss Fundamental Theorem of Algebra.

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