Polynomial and Rational College Algebra Chapter 4 Polynomial and Rational Functions

4.1 Polynomial Long Division and Synthetic Division When the division cannot be completed by factoring, polynomial long division is used and closely resembles whole number division In the division process, zero “place holders” are sometimes used to ensure that like place values will “line up” as we carry out the algorithm

4.1 Polynomial Long Division and Synthetic Division

4.1 Polynomial Long Division and Synthetic Division How to handle a remainder

4.1 Polynomial Long Division and Synthetic Division

The same idea holds for polynomials 4.1 Polynomial Long Division and Synthetic Division If one number divides evenly into another, it must be a factor of the original number The same idea holds for polynomials This means that division can be used as a tool for factoring We need to do two things first Find a more efficient method for division Find divisors that give a remainder of zero

5 1 -2 -13 -17 Synthetic Division 5 15 10 2 -7 1 3 remainder 4.1 Polynomial Long Division and Synthetic Division Synthetic Division 5 1 -2 -13 -17 5 15 10 2 -7 1 3 remainder Multiply in the diagonal direction, add in the vertical direction Explanation of why it works is on pg 376

4.1 Polynomial Long Division and Synthetic Division

4.1 Polynomial Long Division and Synthetic Division

Principal of Factorable Polynomials 4.1 Polynomial Long Division and Synthetic Division Synthetic Division and Factorable Polynomials Principal of Factorable Polynomials Given a polynomial of degree n>1 with integer coefficients and a lead coefficient of 1 or -1, the linear factors of the polynomial must be of the form (x-p) where p is a factor of the constant term. Use synthetic division to help factor Hint: Start with what is easiest 1 1 -10 -4 24

Synthetic Division and Factorable Polynomials 4.1 Polynomial Long Division and Synthetic Division Synthetic Division and Factorable Polynomials

What values of k will make x-3 a factor of 4.1 Polynomial Long Division and Synthetic Division What values of k will make x-3 a factor of

4.1 Polynomial Long Division and Synthetic Division Homework pg 380 1-58

Use the remainder theorem to find the value of H(-5) for 4.2 The Remainder and Factor Theorems The Remainder Theorem If a polynomial P(x) is divided by a linear factor (x-r), the remainder is identical to P(r) – the original function evaluated at r. Use the remainder theorem to find the value of H(-5) for

Use the remainder theorem to find the value of P(1/2) for 4.2 The Remainder and Factor Theorems Use the remainder theorem to find the value of P(1/2) for

The Factor Theorem Given P(x) is a polynomial, 4.2 The Remainder and Factor Theorems The Factor Theorem Given P(x) is a polynomial, If P(r) = 0, then (x-r) is a factor of P(x). If (x-r) is a factor of P(x), then P(r) = 0 Use the factor theorem to find a cubic polynomial with these three roots:

4.2 The Remainder and Factor Theorems A polynomial P with integer coefficients has the zeros and degree indicated. Use the factor theorem to write the function in factored and standard form.

4.2 The Remainder and Factor Theorems Complex numbers, coefficients, and the Remainder and Factor Theorems Show x=2i is a zero of:

Complex Conjugates Theorem 4.2 The Remainder and Factor Theorems Complex Conjugates Theorem Given polynomial P(x) with real number coefficients, complex solutions will occur in conjugate pairs. If a+bi, b≠0, is a solution, then a-bi must also be a solution.

Polynomial zeroes theorem 4.2 The Remainder and Factor Theorems Roots of multiplicity Some equations produce repeated roots. Polynomial zeroes theorem A polynomial equation of degree n has exactly n roots, (real and complex) where roots of multiplicity m are counted m times.

4.2 The Remainder and Factor Theorems Homework pg 389 1-86

The Fundamental Theorem of Algebra 4.3 The Zeroes of Polynomial Functions The Fundamental Theorem of Algebra Every complex polynomial of degree n≥1 has at least one complex root. Our search for a solution will not be fruitless or wasted, solutions for all polynomials exist. The fundamental theorem combined with the factor theorem enables to state the linear factorization theorem.

Linear factorization theorem 4.3 The Zeroes of Polynomial Functions Linear factorization theorem Every complex polynomial of degree n ≥ 1 can be written as the product of a nonzero constant and exactly n linear factors THE IMPACT Every polynomial equation, real or complex, has exactly n roots, counting roots of multiplicity

4.3 The Zeroes of Polynomial Functions Find all zeroes of the complex polynomial C, given x = 1-I is a zero. Then write C in completely factored form:

The Intermediate Value Theorem (IVT) 4.3 The Zeroes of Polynomial Functions The Intermediate Value Theorem (IVT) Given f is a polynomial with real coefficients, if f(a) and f(b) have opposite signs, there is at least one value r between a and b such that f(r)=0 HOW DOES THIS HELP??? Finding factors of polynomials

The Rational Roots Theorem (RRT) 4.3 The Zeroes of Polynomial Functions The Rational Roots Theorem (RRT) Given a real polynomial P(x) with degree n ≥ 1 and integer coefficients, the rational roots of P (if they exist) must be of the form p/q, where p is a factor of the constant term and q is a factor of the lead coefficient (p/q must be written in lowest terms) List the possible rational roots for

4.3 The Zeroes of Polynomial Functions Tests for 1 and -1 If the sum of all coefficients is zero, x = 1 is a rood and (x-1) is a factor. After changing the sign of all terms with odd degree, if the sum of the coefficients is zero, then x = -1 is a root and (x+1) is a factor.

4.3 The Zeroes of Polynomial Functions Homework pg 403 1-106

THE END BEHAVIOR OF A POLYNOMIAL GRAPH 4.4 Graphing Polynomial Functions THE END BEHAVIOR OF A POLYNOMIAL GRAPH If the degree of the polynomial is odd, the ends will point in opposite directions: 1. Positive lead coefficient: down on left, up on right (like y=x3) Negative lead coefficient: up on left, down on right (like y=-x3) If the degree of the polynomial is even, the ends will point in the same direction: Positive lead coefficient: up on left, up on right (like y=x2) Negative lead coefficient: down on left, down on right (like y=-x2)

Attributes of polynomial graphs with roots of multiplicity 4.4 Graphing Polynomial Functions Attributes of polynomial graphs with roots of multiplicity Zeroes of odd multiplicity will “cross through” the x-axis Zeroes of even multiplicity will “bounce” off the x-axis Cross through bounce

Estimate the equation based on the graph 4.4 Graphing Polynomial Functions Estimate the equation based on the graph

g(x) = (x - 2)² (x + 1)³ (x - 1)² 4.4 Graphing Polynomial Functions Estimate the equation based on the graph g(x) = (x - 2)² (x + 1)³ (x - 1)²

Guidelines for Graphing Polynomial Functions Determine the end behavior of the graph Find the y-intercept f(0) = ? Find the x-intercepts using any combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula. Use the y-intercepts, end behavior, the multiplicity of each zero, and a few mid-interval points to sketch a smooth, continuous curve.

Sketch the graph of Down, Down F(0) = -12 4.4 Graphing Polynomial Functions Sketch the graph of Determine the end behavior of the graph Find the y-intercept f(0) = ? Find the x-intercepts using any combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula. Use the y-intercepts, end behavior, the multiplicity of each zero, and a few mid-interval points to sketch a smooth, continuous curve. Down, Down F(0) = -12 bounce Cut through Cut through

f(x) = x⁶ - 2 x⁵ - 4 x⁴ + 8 x³ 4.4 Graphing Polynomial Functions Determine the end behavior of the graph Find the y-intercept f(0) = ? Find the x-intercepts using any combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula. Use the y-intercepts, end behavior, the multiplicity of each zero, and a few mid-interval points to sketch a smooth, continuous curve.

4.4 Graphing Polynomial Functions Homework pg 415 1-86

Vertical Asymptotes of a Rational Function 4.5 Graphing Rational Functions Vertical Asymptotes of a Rational Function Given is a rational function in lowest terms, vertical asymptotes will occur at the real zeroes of g The “cross” and “bounce” concepts used for polynomial graphs can also be applied to rational graphs cross bounce

x- and y-intercepts of a rational function 4.5 Graphing Rational Functions x- and y-intercepts of a rational function Given is in lowest terms, and x = 0 in the domain of r, To find the y-intercept, substitute 0 for x and simplify. If 0 is not in the domain, the function has no y-intercept To find the x-intercept(s), substitute 0 for f(x) and solve. If the equation has no real zeroes, there are no x-intercepts. Determine the x- and y-intercepts for the function

Determine the x- and y-intercepts for the function 4.5 Graphing Rational Functions Determine the x- and y-intercepts for the function Y-intercept No x-intercept

Given is a rational function in lowest 4.5 Graphing Rational Functions Given is a rational function in lowest terms, where the lead term of f is axn and the lead term of g is bxm Polynomial f has degree n, polynomial g has degree m If n<m, the graph of h has a horizontal asymptote at y=0 (the x-axis) If n=m, the graph of h has a horizontal asymptote at y=a/b (the ratio of lead coefficients) If n>m, the graph of h has no horizontal asymptote

Guidelines for graphing rational functions pg 428 Find the y-intercept [evaluate r(0)] Locate vertical asymptotes x=h [solve g(x) = 0] Find the x-intercepts (if any) [solve f(x) = 0] Locate the horizontal asymptote y = k (check degree of numerator and denominator) Determine if the graph will cross the horizontal asymptote [solve r(x) = k from step 4 If needed, compute the value of any “mid-interval” points needed to round-out the graph Draw the asymptotes, plot the intercepts and additional points, and use intervals where r(x) changes sign to complete the graph Given is a rational function in lowest terms, where the lead term of f is axn and the lead term of g is bxm

4.5 Graphing Rational Functions Find the y-intercept [evaluate r(0)] Locate vertical asymptotes x=h [solve g(x) = 0] Find the x-intercepts (if any) [solve f(x) = 0] Locate the horizontal asymptote y = k (check degree of numerator and denominator) Determine if the graph will cross the horizontal asymptote [solve r(x) = k from step 4 If needed, compute the value of any “mid-interval” points needed to round-out the graph Draw the asymptotes, plot the intercepts and additional points, and use intervals where r(x) changes sign to complete the graph

4.5 Graphing Rational Functions Homework pg 431 1-70

Given is a rational function in lowest 4.5 Graphing Rational Functions Given is a rational function in lowest terms, where the lead term of f is axn and the lead term of g is bxm Polynomial f has degree n, polynomial g has degree m If n<m, the graph of h has a horizontal asymptote at y=0 (the x-axis) If n=m, the graph of h has a horizontal asymptote at y=a/b (the ratio of lead coefficients) If n>m, the graph of h has no horizontal asymptote

Oblique and nonlinear asymptotes 4.6 Additional Insights into Rational Functions Oblique and nonlinear asymptotes Given is a rational function in lowest terms, where the degree of f is greater than the degree of g. The graph will have an oblique or nonlinear asymptote as determined by q(x), where q(x) is the quotient of

4.6 Additional Insights into Rational Functions

Choose one application problem 4.6 Additional Insights into Rational Functions Choose one application problem

4.6 Additional Insights into Rational Functions Homework pg 445 1-62

Solving Polynomial Inequalities 4.7 Polynomial and Rational Inequalities – An Analytical View Solving Polynomial Inequalities Given f(x) is a polynomial in standard form pg 452 Use any combination of factoring, tests for 1 and -1, the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero. Plot the zeroes on a number line (x-axis) and determine if the graph crosses (odd multiplicity) or bounces (even multiplicity) at each zero. Recall that complex zeroes from irreducible quadratic factors can be ignored. Use end behavior, the y-intercept, or a test point to determine the sign of the function in a given interval, then label all other intervals as P(x) < 0 or P(x) > 0 by analyzing the multiplicity of neighboring zeroes. State the solution using interval notation, noting strict/non-strict inequalities.

End behavior is down/up 4.7 Polynomial and Rational Inequalities – An Analytical View Use any combination of factoring, tests for 1 and -1, the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero. Plot the zeroes on a number line (x-axis) and determine if the graph crosses (odd multiplicity) or bounces (even multiplicity) at each zero. Recall that complex zeroes from irreducible quadratic factors can be ignored. Use end behavior, the y-intercept, or a test point to determine the sign of the function in a given interval, then label all other intervals as P(x) < 0 or P(x) > 0 by analyzing the multiplicity of neighboring zeroes. State the solution using interval notation, noting strict/non-strict inequalities. Synthetic division cross bounce End behavior is down/up up down f(x) > 0 f(x) < 0 f(x) < 0

Add coefficients 1+1+-5+3=0 4.7 Polynomial and Rational Inequalities – An Analytical View Use any combination of factoring, tests for 1 and -1, the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero. Plot the zeroes on a number line (x-axis) and determine if the graph crosses (odd multiplicity) or bounces (even multiplicity) at each zero. Recall that complex zeroes from irreducible quadratic factors can be ignored. Use end behavior, the y-intercept, or a test point to determine the sign of the function in a given interval, then label all other intervals as P(x) < 0 or P(x) > 0 by analyzing the multiplicity of neighboring zeroes. State the solution using interval notation, noting strict/non-strict inequalities. Test for 1 and -1 Add coefficients 1+1+-5+3=0 Means that x=1 is a root End behavior down/up bounce cross f(x) < 0 f(x) > 0 f(x) > 0

The graph will change signs at x = 2, -3, and 7/4 4.7 Polynomial and Rational Inequalities – An Analytical View The graph will change signs at x = 2, -3, and 7/4 The y-intercept is 7/6 which is positive above above below below

4.7 Polynomial and Rational Inequalities – An Analytical View above above below

4.7 Polynomial and Rational Inequalities – An Analytical View Homework pg 458 1-66

Chapter 4 Review

Chapter 4 Review

Chapter 4 Review Use synthetic division to show that (x+7) is a factor of 2x4+13x3-6x2+9x+14

Factor and state roots of multiplicity Chapter 4 Review Factor and state roots of multiplicity

State an equation for the given graph Chapter 4 Review State an equation for the given graph

State an equation for the given graph Chapter 4 Review State an equation for the given graph

Chapter 4 Review Graph

Divide using long division Chapter 4 Review Trashketball Review Divide using long division

Use synthetic division to divide Chapter 4 Review Trashketball Review Use synthetic division to divide

Show the indicated value is a zero of the function Chapter 4 Review Trashketball Review Show the indicated value is a zero of the function

Show the indicated value is a zero of the function Chapter 4 Review Trashketball Review Show the indicated value is a zero of the function

Find all the zeros of the function Chapter 4 Review Trashketball Review Find all the zeros of the function Real root x=3 Complex roots x=±2i

Find all the zeros of the function Chapter 4 Review Trashketball Review Find all the zeros of the function

Chapter 4 Review Trashketball Review State end behavior, y-intercept, and list the possible rational roots for each function

Chapter 4 Review Trashketball Review State end behavior, y-intercept, and list the possible rational roots for each function

Chapter 4 Review Trashketball Review Sketch the Graph using the degree, end behavior, x- y-intercept, zeroes of multiplicity and midinterval points

Chapter 4 Review Trashketball Review Sketch the Graph using the degree, end behavior, x- y-intercept, zeroes of multiplicity and midinterval points

Graph using guidelines for graphing rational functions Chapter 4 Review Trashketball Review Graph using guidelines for graphing rational functions

Graph using guidelines for graphing rational functions Chapter 4 Review Trashketball Review Graph using guidelines for graphing rational functions