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Block 3 Unit 3
Transcript of Block 3 Unit 3
Polynomial Functions of Higher Degree section 2.3 section 2.4 section 2.5 section B4 Unit #3 By: Ureka Ajawara, Lorene Halford, Cat James,
Sami Armiger, Chloe Winn The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra:
if a function n is bigger than zero, it has at least one complex zero. Linear Factorization Theorem:
If a function n is bigger than zero, it has n linear factors. Ex: Confirming Real Zeros Conjugates Complex Zeros Come in Conjugate Pairs Ex: Finding a Polynomial when Given Zeros Real Zeros of Polynomial Functions Objectives Use long and synthetic division to divide polynomials by other polynomials
Use the remainder and factor theorems
Use the Rational Zeros Test to determine possible rational zeros of a polynomial LONG DIVISION Example: Use long division to divide Polynomial functions are contiuous graphs
polynomial graphs have smooth round curves not sharp jagged curves
degree 1 polynomials are lines
degree 2 polynomials are parabolas Polynomials to the nth degree have the form of: Remember to use place holders
The remainder is written over the original denominator SYNTHETIC DIVISION Example: Use to do the following: a) Use synthetic division to determine if (x+3) is a factor of f(x) Use the x-intercept to divide with
It is only a factor if the remainder is zero b) Evaluate f(-3) using direct substitution THe Leading coefficient test:
when n is odd and the coefficient is postivie:
when n is odd and the coefficient is negative:
when n is even and the coefficient is positive:
when n is even and the coefficient is negative: REMAINDER THEOREM States that: If f(x) is divided by (x-k), then the remainder, r=f(k). RATIONAL ZEROS TEST real zeros of polynomial functions:
x=a is a zero of the function of x
x=a is a solution of the polynomial equation f(x)=0
(x-a) is a factor of the polynomial f(x)
(a,0) is an x-intercept of the graph of f(x) Ex: a) What is the list of all possible rational zeros? Use Rational Zeros Test Try 1 & -1 if you don't have a calculator Repeated Zeros:
polynomial function, that is a factor of
yields a repeated zero x=a of multiplicity k b) Graph on calculator to eliminate some of the possibilities and write a smaller list of possibilities. Window: [-20,20,1,-5,5,1,1] & [-5,5,1,-5,5,1,1]
Possible: -3. -1.5, -1, and 2 c) Use synthetic division to confirm to the zeros (must get remainder of 0) d) Write f(x) in factored form intermediate Value Theorem f(x)=(x-2)(x+1)(x+1.5)(x+3) Solving Inequalities Algebraically and Graphically Properties of Inequalities When each side of an inequalities is multiplied
or divided by a negative number, the direction of the
inequality symbo lmust be reversed in order to
keep a true statement.
Equivilant inequalities: Two inequalities that have the same solution set Ex: Solve 5x-7 >3x+9 5x-7>3x+9
x>8 Ex: Solve -3≤6x-1<3 Note: On a number line, open circles mean that
the number is not included. Darkened circles mean
that the number is included. example problems are on page
169 #13-32 Absolute Value Inequalities Double inequality Compound inequality Example Note: When solving a polynomial
inequality, remember to make the table for
the test intervals using the critical numbers.
This can be found on page A68 More sample problems can be found throughout
pages A63-A74 Operations With Complex Numbers Subtraction Division Graph:
Plot the following Complex Numbers: Section B.4 Can't have
and a no
in there. Objectives:
solve quadratic inequalities algebraically
and graphically. find the degree of f(x)
describe the end behavior Objectives Analyze graphs of quadratic functions
Write quadratic functions in standard form and use this to sketch graphs of functions
Find the min and max, and use this to apply to real-life applications Definition of Quadratic functions Describing Quadratic Graphs To Write a Quadratic Equation in Standard Form... To find x-intercepts....
Take the original equation and reverse FOIL
Use the Quadratic Equation.... Using the Qudratic Equation for real-life applications Suggested Practice Problems!!
-Page 169 #1-12