Section 2.1 Vocabulary Definition of Polynomial function • Let n be a nonnegative integer and let an,an-1, …, a2, a1, a0 be real numbers with an not equal to zero. The function given by F(x) = anxn + an-1xn-1+….. + a2x2 + a1x + a0 Is called a polynomial function in x of degree n. Definition of Quadratic Function • Let a, b, and c be real numbers with a not equal to zero. The function given by F(x) = ax2 + bx + c is called a quadratic function The graph of a quadratic function is a special type of U-shaped curve called a parabola. All parabolas are symmetric with respect to a line called the axis of symmetry. Standard form of a Quadratic Function •The quadratic function given by 2 F(x) = a(x – h) + k , where a cannot equal zero Maximum and Minimum values of Quadratic Functions • If a > 0, f has a minimum value at x = -b / (2a) • If a < 0 , f has a maximum value at x = -b/(2a) Section 2.2 Vocabulary Continuous • In order to be continuous the graph of the polynomial function has no breaks, holes, or gaps. Leading Coefficient Test • If the highest exponent is odd, and the leading coefficient is positive: Falls left, rises right • If the highest exponent is odd, and the leading coefficient is negative: Rises left falls right • If the highest exponent is even, and the leading coefficient is positive: Rises left and right • If the highest exponent is even, and the leading coefficient is negative: Falls left and right A polynomial n anx + …+ a1x + a0 has at most n real zeros, and at most n-1 relative extrema( minima, or maxima) Real Zeros of Polynomial Functions If a is a real number than the following statements are equivalent: • x= a is a real zero of the function • X = a is a solution of the polynomial equation f(x) = 0 • (x-a) is a factor of the polynomial f(x). • (a,0) is an x-intercept of the graph of f(x). 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. 1. If k is even, the graph touches 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). Section 2.3 Vocabulary 2 methods of dividing polynomials •Long Division •Synthetic Division(only when the divisor has the form x – k) Division algorithm • If f(x) and d(x) are polynimials where 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) Note: f(x) / d(x) is improper r(x) / d(x) is proper The remainder theorem •If a polynomial f(x) is divided by x – k, the remainder is r = f(k). The Factor Theorem •A polynomial f(x) has a factor of (x – k) if and only if f(k) = 0 Descartes' Rule of Signs • Let f(x) = anxn + …+ a1x + a0 be a polynomial with real coefficients and a0 ≠ 0. 1. The number of positive real zeros of f is either equal to the number of variations of the sign of f(x) or less than that number by an even integer 2. The number of negative real zeros of f is wither equal to the number of variations of the sign of f(-x) or less than that number by an even integer A variation in sign means that two consecutive (non zero) coefficients have opposite signs. A real number b is an upper bound for the real zeros of f is no zeros are greater than b. Similarly, b is a lower bound if no real zeros of f are less than b. Section 2.4 Vocabulary Complex number Has a real part(a) and an imaginary part (bi) and is written in standard form: a + bi 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 Pairs of complex numbers of the forms a + bi and a – bi are called complex conjugates Section 2.5 Vocabulary 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, there n > 0, f has precisely n linear factors f(x) = an(x – c1) (x – c2) …..( x – cn) Where c1, c2,…., cn are complex numbers. A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Section 2.6 Vocabulary A rational function can be written in the form: f(x) = N(x) / D(x) The line x = a is a vertical asymptote of the graph of f if f(x) approaches infinity or f(x) approaches negative infinity as x approaches a, either from the right or from the left. The line y = b is a horizontal asymptote of the graph of f if f(x) approaches b as x approaches positive or negative infinity. Section 2.7 Vocabulary Consider a rational function. If the degree of the numerator is exactly one more than the degree of the denominator then the function has a slant asymptote.