Polynomial Expressions and Functions

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 1 - of 36 Topic 32: Polynomial Expressions and Functions Recall the definitions of polynomials and terms. Definition: A polynomial is a sum of terms. Ex. 1 Which of the following represent polynomial expressions? Of the expressions that are not polynomials, identify the parts that are not properly polynomial terms. Definition: A polynomial term is a product of constants and variables with non-negative integer exponents. 3 2 2x 6x x 5 x 100 100 x 4 2 3x 2 3 5x 7x 0.5 4 x 2 6x 9 3 2 x x 3 2 x

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 2 - of 36 More Terminology Recall Polynomial Functions and their parts The degree of a term is equal to the sum of the exponents of variables in the term. Definition: A polynomial function is a function defined by a polynomial expression. The degree of a polynomial is equal to the greatest degree for any term in the polynomial. The leading term of a polynomial is the term with the highest degree in a one variable polynomial. The leading coefficient of a polynomial is the coefficient of the leading term. A constant term is a term whose degree is zero and is usually represented by a number. Descending order is an ordering scheme for polynomials of one variable where the terms are arranged by degree from highest to lowest. Ex. 2 Answer the following questions about the polynomial function. 2 4 3 6 2 5 f x x x x a. What is the degree of f? b. What is the constant term of f? c. What is the leading term of f? d. What is the leading coefficient of f? e. Write f in descending order.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 3 - of 36 Ex. 3 Answer the following questions about the polynomial function. 2 4 2 1 2 2 g x x x x What is the simplest 8 th degree polynomial function? a. What is the degree of g? b. What is the constant term of g? How can you generalize an 8 th degree monomial function? c. What is the leading term of g? d. What is the leading coefficient of g? True or False: Every 8 th degree function has a term of degree 5. e. Find g(0). True or False: 8 th degree functions can have a term of degree 9.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 4 - of 36 General Polynomial Function Form How can you generalize an 8 th degree polynomial function? How can you generalize an n th degree polynomial function?

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 5 - of 36 Turning Points of a Polynomial Function Recall that turning points are points where a graph switches from increasing to decreasing, or vice versa, and they are the locations of relative extreme points for the function. Certain rules apply for the relationship between polynomial functions and the number of turning points the function s graph may have. 1. Functions of even degree must have an odd number of turning points with at least one turning point having an absolute extreme value while functions of odd degree have an even number of turning points. 2. A function of n degree may never have more than n 1 turning points. A function with k turning points must have a degree of k + 1 or higher. Ex. 3 What can be said about the polynomial function, based on its turning points, that created the graph below?

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 6 - of 36 Topic 33: Behavior of a Polynomial Function End behavior is the term used to describe what happens to function values as x gets very large positive (x ) or very large negative (x ). With polynomial functions, end behavior is always either increasing without bound (graphically, going up) or decreasing without bound (graphically, going down). The end behavior of a polynomial function is determined by the leading coefficient and the degree of the function. 1. The leading coefficient affects right end behavior (x ). a. If the leading coefficient is positive, then the right end of the graph will go up. b. If the leading coefficient is negative, then the right end of the graph will go down. 2. The function s degree affects left end behavior(x ). a. If the degree of the function is even, then the left end of the graph will having matching behavior to the right end of the graph. b. If the degree of the function is odd, then the left end of the graph will have opposite behavior to the right end of the graph.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 7 - of 36 Ex. 1 Determine the end behavior of f and justify your answer. 3 7 5 4 f x x x x Ex. 2 Determine the end behavior of f and justify your answer. 1 2 g x x x 2 3 4

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 8 - of 36 Zeros and their Multiplicity. Effects of multiplicity on the graph of a function Recall that a zero of a function is a value of the input variable for which the function s value is 0. That is, if x is the independent variable, f(x) = 0. Also recall that the x-intercepts of a function occur at the zeroes of the function. Definition: The multiplicity of a zero is the number of times a zero of a function occurs in the fully factored form. In other words, if a function is factored as k f x x n, so that the only zero of the function is n, we say that the zero n has a multiplicity of k to differentiate it from a zero whose factor is raised to a different exponent. 1. At an x-intercept corresponding to a zero of odd multiplicity the graph of the function will cross the x-axis. 2. At an x-intercept corresponding to a zero of even multiplicity the graph of the function will turn at the x-axis. 3. The higher the multiplicity of a zero, the greater the localized flatness of the graph at the corresponding x-intercept. One more connection with multiplicity and factored form is that the sum of all the multiplicities will equal the degree of the function.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 9 - of 36 Ex. 3 Find the intercepts of each polynomial function and discuss the end behavior. P x x 1 x 1 x 2 Ex. 4 Find the intercepts of each polynomial function, describing the behavior of the graph at each x-intercept, and discuss the end behavior. 1 3 P x x 1 x 2 4 Intercepts: Intercepts: End behavior: End behavior:

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 10 - of 36 Ex. 5 Find the intercepts of each polynomial function, describing the behavior of the graph at each x-intercept, and discuss the end behavior. 3 3 2 P x x x 3 x 4 400 Find a polynomial with the properties given below. Write in factored form. Ex. 6a Degree 3, Leading coefficient 1 Zeros Intercepts: Ex. 6b Degree 4 Zero 2, multiplicity 2 End behavior: Ex. 6c Degree 6 Leading coefficient of 1 8 Zero 0, multiplicity 1 Zero 3, multiplicity 2

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 11 - of 36 Topic 34: Division of Polynomials There are two strategies for dividing polynomials, one which is universally applicable (long division) and one which is not but is far more efficient (synthetic division) Long Division Algorithm for Polynomials 3. Subtract. 4. Repeat steps 1-3 by bringing down terms of the dividend after subtraction. The algorithm is completed once the remaining leading term under the dividend is of lower degree than the degree of the divisor. Px For a division problem in the form ( ), where P Qx ( ) represents the dividend & Q represents the divisor: 1. Divide the leading term of the divisor into the leading term of the dividend. Write in the quotient. 2. Multiply the result of step 1 to the divisor and write under the dividend. The result of division creates a quotient, usually labeled as q, and a remainder r. Notationally we can express this as Px ( ) q x Qx ( ) To check division: r x Q x quotient divisor remainder dividend

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 12 - of 36 Ex. 1 Divide. 3 2 x 5x 2x 3 x 2

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 13 - of 36 Ex. 2 Divide. 4 3 2x 6x 2x 7 x 2 2x 2

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 14 - of 36 Ex. 3 Divide. 3 2 4x 8x 6x 5 2x 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 15 - of 36 Synthetic Division Algorithm for Polynomials Px ( ) For a division problem in the form, x c c an an a a 2 a 1 a 0, set up the problem as where each a is a coefficient of P: 1. Bring the first coefficient down under the second line. 2. Multiply the number under the second line to c. Write the product under the next coefficient. 3. Add and write the sum under the second line. 4. Repeat steps 2 and 3 until the coefficients of P are exhausted. 5. The last number at bottom is the remainder. 6. The preceding numbers at bottom are the coefficients of the quotient which should be exactly one degree less than P. Ex. 4 Divide. 3 2 x 5x 2x 3 x 2

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 16 - of 36 Ex. 5 Divide. Ex. 6 Divide. 4 3 2x 5x 2x 9 x 3 3 2 x 8x 2x x 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 17 - of 36 Topic 35: The Remainder & Factor Theorems The Remainder Theorem states that the remainder found by dividing by x c is equivalent to evaluation of the dividend at c. That is, If r(x) = n when Px ( ), then P(c) = n. x c The Remainder Theorem has significant consequences in computer science but for algebra, a specific case called the Factor Theorem has greater relevance. Px ( ) If r(x) = 0 when, x c x c is a factor of P. then c is a zero of P and Ex. 1 Evaluate f(4) by using synthetic division. 3 2 f x x 3x x 6 Ex. 2a Confirm by synthetic division that 4 is a 3 2 zero of f x x 9x 14x 24 using the Remainder Theorem.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 18 - of 36 The Factor Theorem assists in factorization by finding one factor of a polynomial with the hope that the rest of the expression can be factored by simpler techniques. 3 2 Ex. 2b Factor f x x 9x 14x 24 when given that one factor is x 4. 4 3 2 f x x 2x 5x 6x when given that one factor is x + 2. Ex. 3 Factor

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 19 - of 36 Topic 36: Solving Rational Equations Rational equations have one or more rational expressions in the equation. To solve, find the least common denominator and multiply the LCD to both sides of the equation. Solve the resulting equation and check your results. Ex. 1 Solve. 8 x 1 2 x 2 x 4

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 20 - of 36 Ex. 2 Solve. 7x 11 2 1 x 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 21 - of 36 Ex. 3 Solve. x 2 8 4 2 4x 5 x 5 x 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 22 - of 36 Ex. 4 Solve. 2x 1 8 1 2 x 7x 12 x 3 x 4

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 23 - of 36 Topic 37: Solving Rational Inequalities Process for solving rational inequalities: Solving rational inequalities is similar to solving polynomial inequalities. If a rational inequality involves a rational expression compared to zero, the approach is almost exactly the same as a polynomial inequality. Both the numerator and denominator need to be factored (when appropriate) and the most significant difference is that any boundary created from the denominator must be open regardless of the inequality symbol (since setting the denominator equal to zero cannot produce a solution). When a rational inequality does not involve a comparison to zero, the complexity of the process is increased because it is necessary to rewrite the inequality so that one side is zero. 1. As necessary, rewrite so that one side is zero. 2. As necessary, factor the expression. 3. Set each factor equal to zero and solve to find boundary values for the solution intervals. 4. Draw a number line and mark the boundaries with an appropriate dot (closed if equal to the numerator, open if not equal to or if coming from the denominator). 5. Pick a test value from each interval formed by the boundaries and evaluate the expression using each one. Only the sign of the evaluation is relevant (all positive numbers are greater than zero, all negatives are less than zero.) 6. Identify the solution intervals.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 24 - of 36 Ex. 1 Solve. Present solutions graphically & in interval notation. 2x 3 0 2 x 2x

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 25 - of 36 Ex. 2 Solve. Present solutions graphically & in interval notation. 3 x 1 2 x 4

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 26 - of 36 Ex. 3 Solve. Present solutions graphically & in interval notation. x 4 1 2x 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 27 - of 36 Topic 38: Rational Functions Intercepts & Vertical Asymptotes of Rational Functions Definition: A rational function is defined by the quotient of two polynomial functions. Let R represent a rational function and P and Q P x represent polynomial functions, then R x Q x Definition:. An asymptote is a line that a graph approaches but does not intersect as values appr The y-intercept occurs at R(0). The x-intercept(s) occur wherever P(x) = 0 but Q(x (That is, an x-value makes the numerator equal 0 but not the denominator.) Vertical Asymptotes occur wherever Q(x) = 0 but P(x (That is, an x-value makes the denominator equal 0 but not the numerator.) A vertical asymptote of a rational function is a line x = h where h is a solution to Q(x) = 0 and not P(x) = 0.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 28 - of 36 Holes of a Rational Function Horizontal Asymptotes of Rational Functions A hole in a function is created when the rule of the rational function can be simplified and the denominator causes a point-wise loss of definition. Holes occur wherever P(x) = 0 and Q(x) = 0. (That is, an x-value makes both the numerator and denominator equal 0.) To find the ordered pair coordinates of a hole, set the common factor of P and Q equal to zero and solve. The solution will be the x-value of the hole. To find the y-value of the hole, simplify the rational expression and evaluate the x-value of the hole. Horizontal Asymptotes correspond with end behavior. Unlike polynomial functions which either increase or decrease without bound as x, rational functions may approach a fixed #. To determine if a rational function has a horizontal asymptote, identify the degrees of the numerator and denominator, then: a: If the degree of the numerator > the degree of the denominator, then the rational function does not have a horizontal asymptote. b: If the degree of the numerator < the degree of the denominator, then the rational function has a horizontal asymptote of y = 0. c: If the degree of the numerator = the degree of the denominator, then the rational function has a horizontal asymptote of y = k where k is the ratio of the leading coefficients of P and Q.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 29 - of 36 Ex. 1 Identify the attributes of the rational function and then sketch a graph. Ex. 2 Identify the attributes of the rational function and then sketch a graph. R x 6 2x x 1 R x x x 2 2 4x 12 3x 4 y-intercept: y-intercept: x-intercept(s): x-intercept(s): vertical asymptote(s): vertical asymptote(s): horizontal asymptote: hole: horizontal asymptote: hole:

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 30 - of 36 Ex. 3 Identify the attributes of the rational function and then sketch a graph. Ex. 4 Identify the attributes of the rational function and then sketch a graph. 2 R x x x 2 3x 4 R x x 2 4 x 2 2x 8 y-intercept: y-intercept: x-intercept(s): x-intercept(s): vertical asymptote(s): vertical asymptote(s): horizontal asymptote: hole: horizontal asymptote: hole:

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 31 - of 36 Some rational functions, instead of having a horizontal asymptote, have a slant asymptote. Definition: A slant asymptote, also called an oblique asymptote, is an asymptote that is neither horizontal nor vertical. Slant asymptotes occur whenever the numerator is exactly one degree greater than the denominator. As such it is not possible to have both a horizontal asymptote and a slant asymptote (but it is possible to have neither). The quotient found by dividing the rational expression defines the rule of the slope of the slant asymptote (i.e. y = quotient). Ex. 5 Find the slant asymptote of the rational function and then determine the intercepts and vertical asymptotes. R x 2 2x 4x 6 x 2

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 32 - of 36 Topic 39: Variation In some situations, it is easy to define one or more independent variables that impact a dependent variable. There are three forms of variation scenarios: Direct Variation y kx Inverse Variation y k x Our approach to variation problems will come in three steps: 1. Use the information given to write a general variation equation. 2. Use given data to solve for the constant of proportionality (also called the constant of variation). 3. Merge the general variation equation and the constant of proportionality to make a specific variation equation and solve. Joint Variation y kxz Reminder: All variation equations must include a constant of proportionality k.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 33 - of 36 Write a general variation equation from each statement. Write a general variation equation from each statement. Ex. 1 P is directly proportional to u. Ex. 4 S varies directly as the product of the squares of r and. Ex. 2 M varies inversely as t. Ex. 5 A is proportional to the second power of t and inversely proportional to the cube of x. Ex. 3 h is inversely proportional to the product of a and b.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 34 - of 36 Ex. 6 Write a general variation equation from each statement and then use the data to find the constant of proportionality. Ex. 7 Write a general variation equation from each statement and then use the data to find the constant of proportionality. t is jointly proportional to x and y and inversely proportional to r. When x = 2, y = 3, and r = 12, t = 25. is proportional to a and inversely proportional to the square of b. When a = 54 and b = 3, = 2.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 35 - of 36 Applications of Variation Ex. 8 The intensity of illumination from a light, I, varies inversely as the square of the distance, d, from the light. A particular lamp has an intensity of 1000 candles at 8 yards. What will be the intensity of the lamp at 20 yards? Ex. 9 The pressure of a sample of gas, P, is directly proportional to the temperature, T, and inversely proportional to the volume, V. If 100 L of gas exerts a pressure of 33.2 kpa at 400 K, determine the pressure exerted by the gas if the temperature is raised to 500 K and the volume is reduced to 80 L.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 36 - of 36 Ex. 10 The maximum weight, M, a beam can support is jointly proportional to its width, w, and the square of its height, h, and inversely proportional to its length, l. A beam with dimensions as shown in the picture below at left can support 4800 lbs. If a beam made from the same type of wood has the dimensions as shown in the picture below at right, what is the maximum weight it can support?