Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x) = a n x n 1 +... + a 2 x 2 + a 1 is called a polynomial function in x of degree n. Definition of Quadratic Function: Let a,b, and c be real numbers with a 0. The function given by f (x) = ax 2 + bx + c is called a quadratic function. Standard Form of a Quadratic Function: Let quadratic equation given by f (x) = a(x 2 h) 2 + k, a 0 is in standard form. The graph of f is a parabola whose axis of symmetry is the vertical line x = h and whose vertex is the point (h, k). If a > 0, the parabola opens upward, and if a < 0 the parabola opens downward. Minimum and Maximum Values of Quadratic Functions: 1. If a > 0, f has a minimum value at x = b 2a. 2. If a > 0, f has a maximum value at x = b 2a.
(from 2.2) Leading Coefficient Test: As x moves without bound to the left or to the right, the graph of the polynomial function f (x) = a n x n 1 +...+ a 1, a n 0, eventually rises or falls in the following manner: 1. When n is odd: *If the leading coefficient is positive (a n > 0), the graph falls to the left and rises to the right. as x as x *If the leading coefficient is negative (a n < 0), the graph rises to the left and falls to the right. as x as x 2. When n is even: *If the leading coefficient is positive (a n > 0), the graph rises to the left and to the right. as x as x *If the leading coefficient is negative (a n < 0), the graph falls to the left and to the right. as x as x Real Zeros of Polynomial Functions: If f is a polynomial function and a is a real number, the following statements are equivalent: 1. x = a is a zero of the function f. 2. x = a is a solution of the polynomial equation f (x) = 0. 3. (x - a) is a factor of the polynomial f (x). 4. (a, 0) is an x-intercept of the graph of f.
Repeated Zeros: For a polynomial function, a factor of (x - a) k, k > 1, yields a repeated zero x = a of multiplicity k. 1. If k is odd, the graph crosses the x-axis at x = a. 2. If k is even, the graph touches the x-axis (but does not cross the x-axis) at x = a. Intermediate Value Theorem: Let a and b be real numbers such that a < b. If f is a polynomial function such that f (a) f (b), then in the interval [a,b], f takes on every value between f (a) and f (b). (from 2.3) The Division Algorithm: If f (x) and d(x) are polynomials such that d(x) 0, and the degree of d(x) is less than or equal to the degree of f (x), there exists unique polynomials q(x) and r(x) such that... f (x) = d(x)q(x) + r(x) f (x) is the dividend, d(x) is the divisor q(x) is the quotient, r(x) is the remainder where r(x) = 0 or the degree of r(x) is less than the degree of d(x). If the remainder, r(x), is zero, d(x) divides evenly into f (x). Synthetic Division (of a Cubic Polynomial): To divide ax 3 + bx 2 + cx + d by x - k, use the following pattern. k a b c d ka mk nk a m n r vertical pattern: Add terms (where b - ka = m; c - mk = n; d - nk = r) Intepret the answer as ax 2 + mx + n + diagonal pattern: Multiply by k. r x k The Remainder Theorem: If a polynomial f (x) is divided by x - k, the remainder is r and r = f (k).
The Factor Theorem: A polynomial f (x) has a factor (x - k) if and only if f (k) = 0. Using the Remainder in Synthetic Division: In summary, the remainder r, obtained in the synthetic division of f (x) by x - k, provides the following information. 1. the remainder r gives the value of f at x = k. That is, r = f (k). 2. If r = 0, (x - k) is a factor of f (x). 3. If r = 0, (k,0) is an x-intercept of the graph of f. The Rational Zero Test: If the polynomial f (x) = a n x n 1 +...+ a 2 x 2 + a 1 has integer coefficients, every rational zero of f has the form... Rational zero = p q where p and q have no common factors other than 1, p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n. Descartes s Rule of Signs: Let f (x) = a n x n 1 +... + a 2 x 2 + a 1 be a polynomial with real coefficients and a 0 0. 1. The number of positive real zeros of f is either equal to the number of variations in the sign of f (x) or less than that number by an even integer. 2. The number of negative real zeros of f is either equal to the number of variations in the sign of f ( x) or less than that number by an even integer. Upper and Lower Bounds: Let f (x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f (x) is divided by x - c, using synthetic division. 1. If c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f. 2. If c < 0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f.
(from 2.4) Definition of a Complex Number: If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form. If b = 0, the number a + bi = a is a real number. If b 0, the number a + bi is called an imaginary number. A number of the form bi, where b 0, is called a pure imaginary number. Equality of Complex Numbers: Two complex numbers a + bi and c + di, written in standard form, are equal to each other a + bi = c + di if and only if a = c and b = d. Addition and Subtraction of Complex Numbers: If a + bi and c + di are two complex numbers written in standard form, their sum and difference are defined as follows: Sum: (a + bi) + (c + di) = (a + c) + (b + d)i Difference: (a + bi) (c + di) = (a c) + (b d)i (from 2.5) The Fundamental Theorem of Algebra: If f (x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system. Linear Factorization Theorem: If f (x) is a polynomial of degree n, where n > 0, f has precisely n linear factors f (x) = a n (x c 1 )(x c 2 )...(x c n ) where c 1,c 2,...,c n are complex numbers. Complex Zeros Occur in Conjugate Pairs: Let f (x) be a polynomial function that has real coefficients. If a + bi, where b 0, is a zero of the function, the conjugate a - bi is also a zero of the function. Factors of a Polynomial: Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.
(from 2.6) Definition of Vertical and Horizontal Asymptotes: 1. The line x = a is a vertical asymptote of the graph of f if or as x a, either from the right (x a + ) or from the left (x a ). 2. The line y = b is a horizontal asymptote of the graph of f if f (x) b as x or x. Vertical and Horizontal Asymptotes of a Rational Function: Let f be a rational function f (x) = N(x) D(x) = a nx n 1 +...+ a 1 b m x m + b m 1 x m 1 +...+ b 1 x + b 0 where N(x) and D(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of D(x). 2. The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x). a. If n < m, the graph of f has the line y = 0 (the x-axis) as a horizontal asymptote. b. If n = m, the graph of f has the line y = a n as a horizontal b m asymptote, where a n is the leading coefficient of the numerator and b m is the leading coefficient of the denominator. c. If n > m, the graph of f has no horizontal asymptote.
(from (2.7) Guidelines for Graphing Rational Functions: Let f (x) = N(x), where N(x) and D(x) are polynomials. D(x) 1. Simplify f, if possible. Any restrictions on the domain of f not in the simplified function should be listed. 2. Find and plot the y-intercept (if any) by evaluating f (0). 3. Find the zeros of the numerator (if any) by setting the numerator equal to zero. Then plot the corresponding x-intercepts. 4. Find the zeros of the denominator (if any) by setting the denominator equal to zero. then sketch the corresponding vertical asymptotes using dashed vertical lines and plot the corresponding holes using open circles. 5. Find and sketch any other asymptotes (horizontal or slant) of the graph using dashed lines. 6. Plot at least one point between and one point beyond each x-intercept and vertical asymptote. 7. Use smooth curves to complete the graph between and beyond the vertical asymptotes, excluding any points where f is not defined.