CHAPTER 3: POLYNOMIAL AND RATIONAL FUNCTIONS 3.1 POLYNOMIAL FUNCTIONS AND MODELING Polynomial Function A polynomial function P is given by P( x) an xn an1xn1 an2 xn2 a1x a0 , where the coefficients an , an 1 ,..., a1 , a0 are real numbers and the exponents are whole numbers. o The first nonzero coefficient, an , is called the leading coefficient o The term an xn is called the leading term o The degree of the polynomial function is n The Leading Term Test o The behavior of the graph of a polynomial function as x becomes very large ( x ) or very small ( x ) is referred to as the end behavior of the graph. The leading term determines a graph’s end behavior o The Leading Term Test If an xn is the leading term of a polynomial function, then the behavior of the graph as x or x can be described in one of the four following ways. 1. If n is even, and an 0 : 2. If n is even, and an 0 : 3. If n is odd, and an 0 : 4. If n is odd, and an 0 : Even and Odd Multiplicity If x c , k 1, is a factor of a polynomial function P( x) and x c k k 1 is not a factor and o k is odd, then the graph crosses the x-axis at c, 0 ; o k is even, then the graph is tangent to the x-axis at c, 0 . Polynomial Models o Cubic Regression o Quartic Regression 3.2 GRAPHING POLYNOMIAL FUNCTIONS Graphing Polynomial Functions If P( x) is a polynomial function of degree n, the graph of the function has at most n real zeros, and therefore at most n x-intercepts; at most n 1 turning points. o Steps for Graphing Polynomial Functions 1. Use the leading-term test to determine the end behavior. 2. Find the zeros of the function by solving f ( x) 0. Any real zeros are the first coordinates of the x-intercepts. 3. Use the zeros (x-intercepts) to divide the x-axis into intervals and choose a test point in each interval to determine the sign of all function values in that interval. 4. Find f (0). This gives the y-intercept of the function. 5. If necessary, find additional function values to determine the general shape of the graph and then draw the graph. 6. As a partial check, use the facts that the graph has at most n x-intercepts and at most n 1 turning points. Multiplicity of zeros can also be considered in order to check where the graph crosses or is tangent to the x- axis. We can also check the graph with a graphing calculator. The Intermediate Value Theorem For any polynomial function P( x) with real coefficients, suppose that for a b , P (a ) and P(b) are of opposite signs. Then the function has a real zero between a and b. 3.3 POLYNOMIAL DIVISION; THE REMAINDER AND FACTOR THEOREMS This section teaches us concepts that help to find the exact zeros of polynomial functions with degree three or higher. Consider the function h( x) x3 2 x 2 5 x 6 x 3 x 1 x 2 This gives us the following zeros: x 3 0 x 3 x 1 0 x 1 3, 1, 2 x 2 0 x 2 When you divide one polynomial by another, you obtain a quotient and a remainder. o If the remainder is zero then the divisor is a factor of the dividend. P ( x) d ( x) Q ( x) R ( x) P ( x) : Dividend d ( x) : Divisor Q ( x) : Quotient R ( x) : Remainder Synthetic Division o Consider the following: 4x 3 3x 2 x 7 x 2 . A. 4 x 2 5 x 11 x 2 4 x3 3x 2 x 7 4 x3 8 x 2 5x 2 x 5x 10 x 2 11x 7 11x 22 29 B. 4 5 11 1 2 4 3 1 7 48 5 1 5 10 11 7 11 22 29 C. 2 4 3 1 7 8 10 22 4 5 11 29 The Remainder Theorem If a number c is substituted for x in the polynomial f ( x) , then the result f (c) is the remainder that would be obtained by dividing f ( x) by x c. In other words, if f ( x) ( x c) Q( x) R, then f (c) R. The Factor Theorem For a polynomial f ( x) , if f (c) 0, then x c is a factor of f ( x). Proof: If we divide f ( x) by x c, we obtain a quotient and a remainder, related as follows: f ( x) ( x c) Q( x) f (c). Then if f (c) 0, we have f ( x) ( x c) Q( x), so x c is a factor of f ( x) . 3.4 THEOREMS ABOUT ZEROS OF POLYNOMIAL FUNCTIONS The Fundamental Theorem of Algebra Every polynomial function of degree n, with n 1, has at least one zero in the system of complex numbers. Every polynomial function f of degree n, with n 1, can be factored into n linear factors (not necessarily unique); that is, f ( x) an x c1 x c2 x cn . Finding Polynomials with Given Zeros o If a complex number a bi, b 0, is a zero of a polynomial function f ( x) with real coefficients, then its conjugate, a bi , is also a zero. Example: Find a polynomial function of degree 3, having the zeros 1, 3i , and 3i. Solution: f ( x) an x 1 x 3i x 3i . The number an can be any nonzero number. The simplest function will be obtained if we let an 1. Then we have f ( x) x 1 x 3i x 3i x 1 x 2 9 x3 9 x x 2 9 x3 x 2 9 x 9 Rational Coefficients If a b c , where a and b are rational and c is not a perfect square, is a zero of a polynomial function f ( x) with rational coefficients, then a b c is also a zero. Integer Coefficients and the Rational Zeros Theorem o The Rational Zeros Theorem Let P( x) an xn an1x n1 an2 xn2 a1x a0 , where all the coefficients are integers. Consider a rational number denoted by p , where p and q are relatively prime (having no common q factor besides 1 and -1). If p is a zero of P( x) , then p is a q factor of a0 and q is a factor of an . Example: Given f ( x) 3x 4 11x3 10 x 4 : a) Find the rational zeros and then the other zeros; that is, solve f ( x) 0. b) Factor f ( x) into linear factors. Solution: a) Because the degree of f ( x) is 4, there are at most 4 distinct zeros. The possibilities for p are q possibilities for p 1, 2, 4 : ; possibilities for q 1, 3 1 1 2 2 4 4 possibilities for p : 1, 1, 2, 2, 4, 4, , , , , , q 3 3 3 3 3 3 Now we need to graph the function on a graphing calculator to see which of these possibilities seem to be zeros. When you divide the function by x (1) or x 1 , you’ll find that -1 is a zero. Dividing the quotient 3x3 14 x 2 14 x 4 (obtained from the first division problem 1 1 above) by x will show that is not a 3 3 2 2 zero. Repeating again for shows that 3 3 is a zero. We obtain a quotient of 2 3x 2 12 x 6 . So we have 1 and as the 3 rational zeros and 3x 2 12 x 6 factors as 3 x 2 4 x 2 , which yields 2 2 as the other two zeros. b) Therefore the complete factorization of f ( x) is 2 f ( x) x 1 x 3x 2 12 x 6 3 2 x 1 x 3 x 2 4 x 2 3 2 3 x 1 x x 2 2 x 2 2 3 3.5 RATIONAL FUNCTIONS A rational function is a function f that is a quotient of two p( x) polynomials, that is, f ( x) , where p x and q x are polynomials q( x) and where q x is not the zero polynomial. The domain of f consists of all inputs x for which q( x) 0. Asymptotes o Vertical Asymptotes The line x a is a vertical asymptote for the graph of f if any of the following are true: f ( x) as x a or f ( x) as x a , or f ( x) as x a or f ( x) as x a . Determining Vertical Asymptotes p( x) For a rational function f ( x) , where p( x) and q( x) q ( x ) are polynomials with no common factors other than constants, if a is a zero of the denominator, then the line x a is a vertical asymptote for the graph of the function. Horizontal Asymptotes Determining a Horizontal Asymptote When the numerator and the denominator of a rational function have the same degree, a the line y is the horizontal asymptote, b where a and b are the leading coefficients of the numerator and the denominator, respectively. When the degree of the numerator of a rational function is less than the degree of the denominator, the x-axis, or y 0, is the horizontal asymptote. When the degree of the numerator of a rational function is greater than the degree of the denominator, there is no horizontal asymptote. Oblique, or Slant, Asymptotes Example: Find all asymptotes of 2 x 2 3x 1 f ( x) x2 Solution: 1. Since x 2 0 x 2 , this gives us a vertical asymptote at the line x 2. 2. There are no horizontal asymptotes since the degree of the numerator is greater than the degree of the denominator. 3. Dividing: (2 x 2 3x 1) ( x 2) , we get 1 2 x 1 . Now we see that when x2 1 x or x , 0 and the value of x2 f ( x) 2 x 1. This means that as the absolute value of x becomes very large, the graph of f ( x) gets very close to the graph of y 2 x 1. Thus the line y 2 x 1 is the oblique asymptote. 3.6 POLYNOMIAL AND RATIONAL INEQUALITIES Polynomial Inequalities o To solve a polynomial inequality: 1. Find an equivalent inequality with 0 on one side. 2. Solve the related polynomial equation. 3. Use the solutions to divide the x-axis into intervals. Then select a test value from each interval and determine the polynomial’s sign on the interval. 4. Determine the intervals for which the inequality is satisfied and write interval notation or set-builder notation for the solution set. Include the endpoints of the intervals in the solution set if the inequality symbol is or . Rational Inequalities o To solve a rational inequality: 1. Find an equivalent inequality with 0 on one side. 2. Change the inequality symbol to an equals sign and solve the related equation. 3. Find the values of the variable for which the related rational function is not defined. 4. The numbers found in steps (2) and (3) are called critical values. Use the critical values to divide the x-axis into intervals. Then test an x-value from each interval to determine the function’s sign in that interval. 5. Select the intervals for which the inequality is satisfied and write interval notation or set-builder notation for the solution set. If the inequality symbol is or , then the solutions from step (2) should be included in the solution set. The x-values found in step (3) are never included in the solution set. 3.7 VARIATION AND APPLICATIONS Direct Variation If a situation gives rise to a linear function f ( x) kx or y kx, where k is a positive constant, we say that we have direct variation, or that y varies directly as x, or that y is directly proportional to x. The number k is called the variation constant, or constant of proportionality. Inverse Variation k k or y , where k If a situation gives rise to a linear function f ( x) x x is a positive constant, we say that we have inverse variation, or that y varies inversely as x, or that y is inversely proportional to x. The number k is called the variation constant, or constant of proportionality. Combined Variation y varies directly as the nth power of x if there is some positive constant k such that y kx n . y varies inversely as the nth power of x if there is some positive k constant k such that y n . x y varies jointly as x and z if there is some positive constant k such that y kxz.