PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011
1 Chapter 5 Section 5.1: Polynomial Functions and Models Section 5.2: Properties of Rational Functions Section 5.3: The Graph of a Rational Function Section 5.4: Polynomial and Rational Inequalities Section 5.5: The Real Zeros of a Polynomial Function Section 5.6: Complex Zeros; Fundamental Theorem of Algebra
Section 5.1: Polynomial Functions and Models Definition Example Determine which of the following are polynomial functions. For those that are, state the degree. (a) f (x) = 2 3x 4 (b) g(x) = x (c) h(x) = 0 (d) s(x) = x2 2 x 3 1 (e) r(x) = 0 (f) t(x) = 2x 3 (x 1) 2
Example Graphs of polynomial functions must be smooth and continuous.
Definition
Note
Example
Definition
Example
Roots of Polynomials Definition
Example Recall solving a quadratic equations by factoring. When you factor this gives you the solutions. Solve f (x) = x 2 x 20 by factoring. What do your solutions mean graphically? Example Solve f (x) = x 2 6x + 9 by factoring. What do your solutions mean graphically?
Example Give me a polynomial with roots of -3, 0, and 4. Is that the only possible answer?
Theorem The ends of a polynomial function resemble that of power functions of the same degree.
Definition If (x r) m is a factor of a polynomial, then we say r is a root of multiplicity m. When graphing polynomial functions you should 1 Find the x (roots) and y intercepts. 2 Pick a value in the interval between the different intercepts and evaluate them. 3 Connect these points with a smooth curve.
Example Graph the polynomial function f (x) = x(x 2) 2 (x + 1) 3.
Note
Example Tell me what you can about the polynomial function f (x) = 2(x + 5)(x 6) 4.
Section 5.2: Properties of Rational Functions Definition A rational function is a function of the form f (x) = p(x) q(x) where p(x) and q(x) are polynomial functions and q(x) is not the zero function. Note: That since it is a fraction that we must watch for values that make the denominator undefined.
Example Find the domain of each rational function. 1 R(x) = 2x2 4 x+5 2 R(x) = 1 x 2 4 3 R(x) = x2 1 x 1
Asymptotes Note Due to having polynomials in the denominator we often have horizontal and vertical asymptotes when graphing rational functions. Recall that an asymptote is an imaginary boundary that the graph gets close to but never quite crosses. Figure: Graph of f (x) = 1 x 2 Figure: Graph of f (x) = 1 (x 2) 3 When graphing on the TI you will often get a line where the asymptote should be.
Locating Vertical Asymptotes Note A rational function R(x), in lowest terms, will have a vertical asymptote at x = r if r is a factor of the denominator q(x). When we say lowest terms we mean that the rational function has been simplified. Example Find the vertical asymptotes. 1 R(x) = x x 2 4
Example Find the vertical asymptotes. 1 R(x) = x2 x 2 +1 2 R(x) = x+3 x 1 3 R(x) = x2 9 x 2 +4x 21
Locating Horizontal and Oblique Asymptotes Definition A rational function R(x) is called proper if the degree of the numerator is less than the degree of the denominator. Otherwise we refer to the rational function as improper. Note When a rational function is proper then it has a horizontal asymptote at y = 0. Figure: Graph of f (x) = 1
If a rational function is improper we must divide the two rational functions to find the remainder. We then analyze what happens as x and x. Example Find all of the asymptotes for f (x) = 3x 4 x 2 x 3 x 2 + 1
Example Find all of the asymptotes for f (x) = 8x 2 x + 2 4x 2 1
Example Find all of the asymptotes for f (x) = 3x 4 + 4 x 3 + 3x
Section 5.3: The Graph of a Rational Function Example Graph f (x) = x 1 x 2 4
Example Graph f (x) = x 2 1 x
Example Graph f (x) = x 4 + 1 x 2
Example Graph f (x) = x 2 1 x
Example Find a possible equation for the following graph.
Section 5.4: Polynomial and Rational Inequalities
Example Solve the inequality x 4 4x 2, and graph the solution set. Step 1: Bring all the terms to the left so that zero is on the right. Step 2: Find the zeros (roots) of f. Step 3: Use the numbers found in 2 to separate the number line into intervals. Step 4: Select a number in each interval and evaluate f at that number.
Example Solve the inequality (x + 3)(4x 2 4x + 1) 0, and graph the solution set.
Solving a Rational Inequality Example Solve the inequality (x+3)(2 x) > 0, and graph the solution set. (x 1) 2 Step 1: Bring all the terms to the left so that zero is on the right. Step 2: Find the zeros (roots) of f. Step 3: Use the numbers found in 2 to separate the number line into intervals. Step 4: Select a number in each interval and evaluate f at that number.
Example Solve the inequality 4x+5 x+2 3, and graph the solution set.
Example Solve the inequality x 6 x+8 < 0, and graph the solution set.
Section 5.5: The Real Zeros of a Polynomial Function Theorem
Long Division vs. Synthetic Division
Long Division vs. Synthetic Division
Theorem Let f be a polynomial function. If f (x) is divided by x c, then the remainder is f (c). Example Find the remainder if f (x) = x 3 + 3x 2 + 2x 1 is divided by (a) x + 2 (b) x 1
Theorem Let f be a polynomial function. Then x c is a factor of f (x) if and only if f (c) = 0. Example Determine if the following are factors of f (x) = 2x 3 x 2 + 4x + 3 (a) x + 1 (b) x 1
Theorem A polynomial function of degree n (n 1) has at most n real zeros.
Example Find the number of possible zeros of f (x) = 3x 6 4x 4 + 3x 3 + 2x 2 x 3.
Example Find the number of possible zeros of f (x) = x 3 8x 2 + 18x 11.
Theorem
Example Use the Rational Root Theorem to find the possible rational zeros of f (x) = 3x 6 4x 4 + 3x 3 + 2x 2 x 3.
Example Use the Rational Root Theorem to find the possible rational zeros of f (x) = x 3 8x 2 + 18x 11.
Example Find the real zeros of f (x) = 2x 3 4x 2 8x 12. Use the factors you found to rewrite the polynomial in factored form.
Example Find the real zeros of f (x) = 2x 4 + 13x 3 + 29x 2 + 27x + 9. Use the factors you found to rewrite the polynomial in factored form.
Theorem Every polynomial with real coefficients can be uniquely factored into a product of linear factors and/or irreducible quadratic functions. Theorem A polynomial with real coefficients of odd degree has at least one real zero. This is because any non-real zeros must come in pairs.
Bounds Theorem
Example Find a bound to the zeros of each polynomial. (a) f (x) = x 5 + 3x 3 9x 2 + 5 (b) g(x) = 4x 5 2x 3 + 2x 2 + 1
Intermediate Value Theorem Theorem Let f denote a polynomial function. If a < b and f (a) and f (b) are of opposite sign, then there is at least one real zero of f between a and b.
Example Show that f (x) = x 5 x 3 1 has a zero between 1 and 2.
Summary Example Use all these resources to solve 3x 3 + 5x 2 + x 3 < 0.
Section 5.6: Complex Zeros; Fundamental Theorem of Algebra Definition Theorem Every complex polynomial function f (x) of degree n 1 has at least one complex zero.
Theorem Theorem Let f (x) be a polynomial whose coefficients are real numbers. If r = a + bi is a zero of f, the complex conjugate r = a bi is a zero of f (i.e. they always come in pairs).
Example You are told a polynomial of degree 5 whose coefficients are real numbers has the zeros 2, 3i, and 2 + 4i. (a) Find the remaining two zeros. (b) Use this info to write the function as a product of linear factors and/or irreducible quadratic functions.
Example You are told a polynomial of degree 4 whose coefficients are real numbers has the zeros 1, 1, and 4 + i. (a) Find the remaining two zeros. (b) Use this info to write the function as a product of linear factors and/or irreducible quadratic functions.
Summary Example Use all these resources to factor x 4 + 2x 3 + x 2 8x 20.