F(x)=x 2 + 2x Degree? How many zeros What are the zeros?

Slides:

Advertisements

Similar presentations

POLYNOMIAL PATTERNS Learning Task: (nomial means name or term.)

Advertisements

Power Functions and Models; Polynomial Functions and Models February 8, 2007.

Do Now Factor completely and solve. x2 - 15x + 50 = 0

7/16/ The Factor Theorem. 7/16/ Factor Theorem Factor Theorem: For a polynomial f(x) a number c is a solution to f(x) = 0 iff (x – c)

Algebraic Fractions and Rational Equations. In this discussion, we will look at examples of simplifying Algebraic Fractions using the 4 rules of fractions.

Section 3.2 Polynomial Functions and Their Graphs.

5-1 Transforming Quadratics in Function Notation and Converting to Standard Form.

Algebra 2 Tri-B Review!. LogarithmsSolving Polynomials Compostion of Functions & Inverses Rational Expressions Miscellaneous

Section 2.3: Polynomial Functions of Higher Degree with Modeling April 6, 2015.

Multiplying Polynomials Factoring Cubes. Multiplying Polynomials What to do…FOIL May have to foil more than once.

Warm Up: Solve & Sketch the graph:. Graphing Polynomials & Finding a Polynomial Function.

Section 3.7 Proper Rational Functions Section 3.7 Proper Rational Functions.

Example Solution Think of FOIL in reverse. (x + )(x + ) We need 2 constant terms that have a product of 12 and a sum of 7. We list some pairs of numbers.

Warm-Up 5/3/13 Homework: Review (due Mon) HW 3.3A #1-15 odds (due Tues) Find the zeros and tell if the graph touches or crosses the x-axis. Tell.

2nd Degree Polynomial Function

8.3 Multiplying Binomials

6.2 Polynomials and Linear Functions

Real Zeros of Polynomial Functions Long Division and Synthetic Division.

Transforming Polynomial Functions. Recall Vertex Form of Quadratic Functions: a h K The same is true for any polynomial function of degree “n”!

Warm Up Foil (3x+7)(x-1) Factors, Roots and Zeros.

Review Operations with Polynomials December 9, 2010.

3.4 Zeros of Polynomial Functions Obj: Find x-intercepts of a polynomial Review of zeros A ZERO is an X-INTERCEPT Multiple Zeros the zeros are x = 5 (mult.

Complex Numbers, Division of Polynomials & Roots.

6-2 Polynomials and Linear Factors. Standard and Factored Form  Standard form means to write it as a simplified (multiplied out) polynomial starting.

Sec 3.1 Polynomials and Their Graphs (std MA 6.0) Objectives: To determine left and right end behavior of a polynomial. To find the zeros of a polynomial.

FACTOR to SOLVE 1. X 2 – 4x X 2 – 17x + 52 (x-10)(x + 6) x = 10, -6 (x-4)(x - 13) x = 4,13.

3.3 (1) Zeros of Polynomials Multiplicities, zeros, and factors, Oh my.

Graphs of polynomials a n > 0a n < 0 n even n odd f(x)=x 3 +2x 2 -x-2g(x)=-x 4 +5x 2 -4 degree 3degree 4 degree n h(x)=a n x n +a n-1 x n-1 +…+a 1 x+a.

Solving equations with polynomials – part 2. n² -7n -30 = 0 ( )( )n n 1 · 30 2 · 15 3 · 10 5 · n + 3 = 0 n – 10 = n = -3n = 10 =

5.2 Polynomials, Linear Factors, and Zeros P

Polynomials of Higher Degree 2-2. Polynomials and Their Graphs  Polynomials will always be continuous  Polynomials will always have smooth turns.

3.2-1 Vertex Form of Quadratics. Recall… A quadratic equation is an equation/function of the form f(x) = ax 2 + bx + c.

Solving Equations Binomials Simplifying Polynomials

Do Now Given the polynomial 8 -17x x - 20x 4 – Write the polynomial in standard form: _______________________________________ – Classify the polynomial.

Solving Polynomials.

7.5 Roots and Zeros Objectives:

Today in Pre-Calculus Notes: –Fundamental Theorem of Algebra –Complex Zeros Homework Go over quiz.

9.3 Factoring Trinomomials. Rainbow Method (2 nd degree Polynomial) 2.

Standard form of an equation means we write an equation based on its exponents. 1.Arrange the terms so that the largest exponent comes first, then the.

Section 4.1 Polynomial Functions and Models.

6.7 Pg.366 This ppt includes 7 slides consisting of a Review and 3 examples.

For each polynomials, follow all the steps indicated below to graph them: (a) Find the x- and y-intercepts of f. (b) Determine whether the graph of f crosses.

Last Answer LETTER I h(x) = 3x 4 – 8x Last Answer LETTER R Without graphing, solve this polynomial: y = x 3 – 12x x.

Section 3.2 Polynomial Functions and Their Graphs

Where am I now? Review for quiz 4.1.

Polynomial Graphs: Zeroes and Multiplicity

Section 3.4 Zeros of Polynomial Functions

3.3 Quadratic Functions Quadratic Function: 2nd degree polynomial

Warm Up Evaluate. 1. –24 2. (–24) Simplify each expression.

Warmup 3-7(1) For 1-4 below, describe the end behavior of the function. -12x4 + 9x2 - 17x3 + 20x x4 + 38x5 + 29x2 - 12x3 Left: as x -,

4.3: Real Zeroes of Polynomials Functions

Do Now Given the polynomial 8 -17x3+ 16x - 20x4 - 8

Mod. 3 Day 4 Graphs of Functions.

2.6 Find Rational Zeros pg. 128 What is the rational zero theorem?

Smooth, Continuous Graphs

Section 3.2 Polynomial Functions and Their Graphs

Section 3.4 Zeros of Polynomial Functions

Page 248 1) Yes; Degree: 3, LC: 1, Constant: 1 3)

Unit 4 Polynomials.

Warm Up - September 27, 2017 Classify each polynomial by degree and by number of terms. 1. 5x x2 + 4x - 2 Write each polynomial in standard form.

Polynomial Multiplicity

Daily Warm-Up: Find the derivative of the following functions

Polynomial Functions and Their Graphs

ZERO AND NEGATIVE EXPONENT Unit 8 Lesson 3

Section 2.3 Polynomial Functions and Their Graphs

End Behaviors Number of Turns

Section 3.2 Polynomial Functions and Their Graphs

DO NOW 5/12/15 Identify the degree and leading coefficient of the following polynomials y = 3x3 – 2x + 7 y = -x4 + 3x2 – x + 8 y = 2x2 – 7x y = -x7 + x4.

How many solutions does the equations have?

Polynomial Functions of Higher Degree

Presentation transcript:

f(x)=x 2 + 2x Degree? How many zeros What are the zeros?

f(x)=x 2 + 2x Degree? How many zeros What are the zeros? 2 2 nd x = -2, x = 0

h(x)= x 3 - x Degree? How many zeros? What are the zeros?

h(x)= x 3 - x Degree? How many zeros? What are the zeros? 3 rd 3 x = -1, 0, 1

j(x) = -x 3 +2x 2 +3x Degree? How many zeros? What are the zeros?

j(x) = -x 3 +2x 2 +3x Degree? How many zeros? What are the zeros? 3 rd 3 x = -1, 0, 3

k(x)= x 4 -5x Degree? How many zeros? What are the zeros?

k(x)= x 4 -5x Degree? How many zeros? What are the zeros? 4 th 4 x = -2, -1, 1, 2

l(x) = -(x 4 -5x 2 + 4) Degree? How many zeros? What are the zeros?

l(x) = -(x 4 -5x 2 + 4) Degree? How many zeros? What are the zeros? 4 th 4 x = -2, -1, 1, 2

m(x)=1/2(x 5 +x 4 -7x 3 -22x x) Degree? How many zeros? What are the zeros?

m(x)=1/2(x 5 +x 4 -7x 3 -22x x) Degree? How many zeros? What are the zeros? 5 th 5 x = -3, -2, 0, 1, 2

n(x)=-1/2(x 5 +x 4 -7x 3 -22x x) What are the zeroes? How many zeroes? Degree?

n(x)=-1/2(x 5 +x 4 -7x 3 -22x x) What are the zeros? How many zeros? Degree? 5 th 5 x = -4, -3, 0, 1, 2

Zeros and degree What is the relationship between degree and number of zeros?

Zeros and degree What is the relationship between degree and number of zeros? They are the same.

Zeroes and degree What is the relationship between degree and number of zeros? Graph the following polynomial: ◦y=(x-1)(x-3)  x-intercepts?  Zeroes?  Number of zeroes?  Degree? y

(x-1)(x-3)=x 2 –x-3x+3 = x 2 -4x+3 Factored form To find the zeros, set each factor =0 Standard form (x-1)=0 X=1 (x-3)=0 X=3

Write the standard form of the equation for the polynomial function with the given zeros Steps: Example: Use the zeroes to write the factors: (x - the zero) Write as a function in factored form. Multiply (Foil) and simplify 0, 4 and -2 (x-0)(x-4)(x+2) f(x) = x(x-4)(x+2) f(x) = x(x 2 +2x-4x-8) f(x)= x(x 2 - 2x - 8) f(x) = x 3 - 2x 2 – 8x

Write the standard form of the equation for the polynomial function with the given zeros 1, 3(multiplicity 2) f(x) = (x-1)(x-3)(x-3) f(x) = (x 2 -3x – x +3)(x-3) f(x) = (x 2 - 4x + 3)(x–3) f(x) = x 3 - 3x 2 - 4x x +3x– 9 f(x) = x 3 - 7x 2 +15x-9 Multiplicity means that a zero is used as a factor more than once

For each function, determine the zeros and their multiplicity 1. y=(x-3)(x+2) 2 2. y=x(x-5) 10 (x+4) 2

For each function, determine the zeros and their multiplicity 1. y=(x-3)(x+2) 2 2. y=x(x-5) 10 (x+4) 2 x = 3 (mult. 1) x = -2 (mult. 2) x = 0 (mult. 1) x = 5 (mult. 10) x = -4 (mult. 2)