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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 Find the quotientof 8x3 27 and 2x 3 4.1 Polynomial Long Division and Synthetic Division Find the quotientof 8x 27 and 2x 3 3 2 x 3 8 x 27 3 2 x 3 8x 0 x 0 x 27 3 2 4.1 Polynomial Long Division and Synthetic Division n 2 19n 2n 4 n 3 3 n 3 2n3 n 2 19n 4 4.1 Polynomial Long Division and Synthetic Division x 4 x 21 x 3 2 x 3 x 4 x 21 2 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 a. Find a more efficient method for division b. Find divisors that give a remainder of zero 4.1 Polynomial Long Division and Synthetic Division Synthetic Division x 3 5 2x 13x 17 x 5 2 1 1 -2 -13 -17 5 15 10 3 2 -7 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 Synthetic Division x 12x 34x 7 x7 3 2 4.1 Polynomial Long Division and Synthetic Division Synthetic Division x 15x 12 x 3 3 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. Hint: Start with what is easiest Use synthetic division to help factor x x 10x 4 x 24 4 3 1 2 1 -10 -4 24 1 24 2 12 3 8 4 6 4.1 Polynomial Long Division and Synthetic Division Synthetic Division and Factorable Polynomials x 4x x 6 3 2 4.1 Polynomial Long Division and Synthetic Division What values of k will make x-3 a factor of x 2 kx 27 4.1 Polynomial Long Division and Synthetic Division Homework pg 380 1-58 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 H x x 3x 8x 5x 6 4 3 2 4.2 The Remainder and Factor Theorems Use the remainder theorem to find the value of P(1/2) for Px x3 2x 2 3x 2 4.2 The Remainder and Factor Theorems The Factor Theorem Given P(x) is a polynomial, 1. If P(r) = 0, then (x-r) is a factor of P(x). 2. 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: x 3, x 2, x 2 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. x 7 , x 7 , x 3, x 1; degree 4 4.2 The Remainder and Factor Theorems Complex numbers, coefficients, and the Remainder and Factor Theorems Show x=2i is a zero of: Px x 3x 4 x 12 3 2 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. 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 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. 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: Cx x 1 2i x 5 i x 6 6i 3 2 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 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 3x 4 14x3 x 2 42x 24 0 4.3 The Zeroes of Polynomial Functions Tests for 1 and -1 1. If the sum of all coefficients is zero, x = 1 is a rood and (x-1) is a factor. 2. 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 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) 2. 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: 1. Positive lead coefficient: up on left, up on right (like y=x2) 2. Negative lead coefficient: down on left, down on right (like y=-x2) 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 f x x 3 x 2 2 Cross through bounce 4.4 Graphing Polynomial Functions Estimate the equation based on the graph g(x) = (x - 2)² (x + 1)³ 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 Guidelines for Graphing Polynomial Functions 1. Determine the end behavior of the graph 2. Find the y-intercept f(0) = ? 3. Find the x-intercepts using any combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula. 4. 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 Sketch the graph of g x x 9x 4x 12 4 2 1. 2. 3. 4. 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 midinterval points to sketch a smooth, continuous curve. Down, Down F(0) = -12 f x 1x 1x 2 x 3 2 bounce Cut through Cut through 4.4 Graphing Polynomial Functions f(x) = x⁶ - 2 x⁵ - 4 x⁴ + 8 x³ 1. 2. 3. 4. 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 midinterval points to sketch a smooth, continuous curve. 4.4 Graphing Polynomial Functions Homework pg 415 1-86 4.5 Graphing Rational Functions Vertical Asymptotes of a Rational Function Given r x f x g x is a rational function in lowest terms, vertical asymptotes will occur at the real zeroes of g 1 f x x2 cross The “cross” and “bounce” concepts used for polynomial graphs can also be applied to rational graphs g x 1 x 22 bounce 4.5 Graphing Rational Functions Given r x x- and y-intercepts of a rational function f x is in lowest terms, and x = 0 in the domain of r, g x 1. To find the y-intercept, substitute 0 for x and simplify. If 0 is not in the domain, the function has no y-intercept 2. 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 2 0 h0 2 0 3 0 10 x2 h x 2 x 3x 10 x2 0 2 x 3x 10 h0 0 0 x2 0,0 0,0 4.5 Graphing Rational Functions Determine the x- and y-intercepts for the function h0 3 02 1 h0 3 Y-intercept 0,3 3 h x 2 x 1 0 3 x2 1 03 No x-intercept 4.5 Graphing Rational Functions Given r x f x is a rational function in lowest g x 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 1. If n<m, the graph of h has a horizontal asymptote at y=0 (the x-axis) 2. If n=m, the graph of h has a horizontal asymptote at y=a/b (the ratio of lead coefficients) 3. If n>m, the graph of h has no horizontal asymptote 3x r x 2 x 2 3x 2 r x 2 x 2 3x 3 r x 2 x 2 4.5 Graphing Rational Functions Guidelines for graphing rational functions pg 428 Given r x f x is a rational function in lowest g x terms, where the lead term of f is axn and the lead term of g is bxm 1. 2. 3. 4. 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) 5. Determine if the graph will cross the horizontal asymptote [solve r(x) = k from step 4 6. If needed, compute the value of any “mid-interval” points needed to round-out the graph 7. 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 3x 2 6 x 3 r x x2 7 1. 2. 3. 4. 5. 6. 7. 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 4.5 Graphing Rational Functions Given r x f x is a rational function in lowest g x 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 1. If n<m, the graph of h has a horizontal asymptote at y=0 (the x-axis) 2. If n=m, the graph of h has a horizontal asymptote at y=a/b (the ratio of lead coefficients) 3. If n>m, the graph of h has no horizontal asymptote 3x r x 2 x 2 3x 2 r x 2 x 2 3x 3 r x 2 x 2 4.6 Additional Insights into Rational Functions Oblique and nonlinear asymptotes f x r x Given is a rational function in lowest g x 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 f q(x), where q(x) is the quotient of g x r x x 1 x 2 x4 1 r x 2 x x2 1 x x x4 1 2 2 x x 1 x x 1 x 2 x q x x q x x 2 2 4.6 Additional Insights into Rational Functions x3 4 x v x 2 x 1 4.6 Additional Insights into Rational Functions Choose one application problem 4.6 Additional Insights into Rational Functions Homework pg 445 1-62 4.7 Polynomial and Rational Inequalities – An Analytical View Solving Polynomial Inequalities Given f(x) is a polynomial in standard form pg 452 1. 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. 2. 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. 3. 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. 4. State the solution using interval notation, noting strict/non-strict inequalities. 4.7 Polynomial and Rational Inequalities – An Analytical View f x x 4x 3x 18, 3 2 f x 0 1. 2. Synthetic division x 2x2 6x 9 x 2x 32 cross 3. 4. 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/nonstrict inequalities. x ,3 3,2 bounce End behavior is down/up up down f(x) < 0 f(x) < 0 f(x) > 0 4.7 Polynomial and Rational Inequalities – An Analytical View x x 5x 3 0 3 2 1. 2. Test for 1 and -1 3. Add coefficients 1+1+-5+3=0 Means that x=1 is a root x 1x 2 2x 3 x 12 x 3 4. 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/nonstrict inequalities. x ,3 End behavior down/up bounce cross f(x) < 0 f(x) > 0 f(x) > 0 4.7 Polynomial and Rational Inequalities – An Analytical View x x 1 x 2 x2 x3 x 1 x 2 0 x2 x3 2 The graph will change signs at x = 2, -3, and 7/4 The y-intercept is 7/6 which is positive 4x 3 x2 4 0 x 2x 3 7 x 3, 2, 4 4x 7 0 x 2x 3 above above below below 4.7 Polynomial and Rational Inequalities – An Analytical View above below above 4.7 Polynomial and Rational Inequalities – An Analytical View Homework pg 458 1-66 Chapter 4 Review x 2x 4 2 x x 3 3x 4 x 1 2 x x Chapter 4 Review Chapter 4 Review Use synthetic division to show that (x+7) is a factor of 2x4+13x3-6x2+9x+14 Chapter 4 Review Factor and state roots of multiplicity hx x 6x 8x 6x 9 4 3 2 Chapter 4 Review State an equation for the given graph f x x 1 x 1x 3 2 f x x 4 2 x 3 4 x 2 2 x 3 Chapter 4 Review State an equation for the given graph x2 4x f x 2 x 4 Chapter 4 Review Graph x2 9 r x 2 x 3x 4 Chapter 4 Review Trashketball Review Divide using long division 2x x 8 2 x 2x x 4 x 5x 6 x2 9x 8 2x 4 2 x 2x x 6x 7; R 8 3 3 2 2 Chapter 4 Review Trashketball Review Use synthetic division to divide x 4 x 5x 6 x2 3 2 x 6x 7; R 8 2 2 x 4 13x 3 6 x 2 9 x 14 x7 2x x x 2 3 2 Chapter 4 Review Trashketball Review Show the indicated value is a zero of the function 1 x ; P x 4 x 3 8 x 2 3x 1 2 Chapter 4 Review Trashketball Review Show the indicated value is a zero of the function x 3i; Px x 2x 9x 18 3 2 Chapter 4 Review Trashketball Review Find all the zeros of the function Real root x=3 Complex roots x=±2i 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- yintercept, zeroes of multiplicity and midinterval points Chapter 4 Review Trashketball Review Sketch the Graph using the degree, end behavior, x- yintercept, zeroes of multiplicity and midinterval points Chapter 4 Review Trashketball Review Graph using guidelines for graphing rational functions Chapter 4 Review Trashketball Review Graph using guidelines for graphing rational functions