Polynomial Functions Polynomial Functions i. Define a polynomial f x an xn an1xn1 ... a1x a0 where an , an 1 ,...a0 are real numbers with an 0, and n is a nonnegative integer. ii. Degree of Polynomial iii. Leading Coefficient iv. Special Names 1. Linear Function: First degree polynomial 2. Quadratic Function: Second degree polynomial 3. Cubic Function: Third degree polynomial 4. Quartic Function: Fourth degree polynomial v. Divide polynomials 1. Synthetic Division 2. Division Algorithm a. If a polynomial f x is divided by a nonzero h x , then there is a quotient polynomial q x and a remainder r x such that f x h x g x r x where either r x 0 or r x has degree less than the degree of the divisor, h x . b. Remainder and Factors: if the remainder is 0 when one polynomial is divided by another polynomial, the divisor and the quotient are factors of the dividend. vi. Apply the Remainder Theorem 1. Remainder Theorem If a polynomial f x is divided by x c, then the remainder is f c . 2. Factor Theorem A polynomial function f x has a linear factor x a if and only if f a 0. vii. Determine the maximum number of zeros of a polynomial A polynomial of degree n has at most n distinct real zeros. Real Zeros i. Rational Zero Test If a rational number r is a zero of the polynomial function, then s 1. r is a factor of the constant term a0 , and 2. s is a factor of the leading coefficient an . ii. Zeros and the Factor Theorem Once some zeros of a polynomial have been found, the Factor Theorem can be used to factor the polynomial, which may lead to additional zeros. iii. Factoring a Polynomial Completely Example: Factor f x 2x5 10x4 7 x3 13x2 3x 9 completely. iv. Finding all Real Zeros 1. Use the Rational Zero Test to find all the rational zeros of the function. 2. Write f x as the product of linear factors, one for each rational zero, and another factor g x . 3. If g x has degree 2, find its zeros by factoring or by using the quadratic formula. 4. If g x has degree 3 or higher, use the Bounds Test, if possible to find lower and upper bounds for the zeros of g. Approximate the remaining zeros graphically. iv. Upper and Lower Bound If all the real zeros of the polynomial of f x are between m and n, m n, then m is called a lower bound and n is called an upper bound for the real zeros of f x . v. Bounds Test Let f x be a polynomial with positive leading coefficient. 1. If d 0 and every number in the last row in the synthetic division of f x by x d is nonnegative, then d is an upper bound for the real zeros of f. 2. If c 0 and the number in the last row of the synthetic division of f x by x c are alternately positive and negative, with 0 considered as either, then c is a lower bound for the real zeroes of f. Graphs of Polynomial Functions Properties of General Polynomial Functions ( f x ax n ) i. Continuity o an unbroken curve with no jumps, gaps, or holes o no sharp corners o Examples of functions that are NOT continuous: i. ii. ii. End Behavior a. Odd b. Even iii. Intercepts a. Polynomial in general form f x an xn an1xn1 ... a1x a0 b. Graph has one y-intercept, which is equal to the constant term, a0 . c. Graph has at most n x-intercepts, which are the real zeros of f. iv. Multiplicity a. If x r is a factor that occurs m times in the complete factorization of a polynomial expression, then r is call a zero with multiplicity m of the related polynomial function. b. Let c be a zero of multiplicity k of a polynomial f. i. If k is odd, the graph of f crosses the x-axis at c. ii. If k is even, the graph of f touches, but does not cross, the xaxis at c. v. Local Extrema A polynomial function of degree n has at most n 1 local extrema. vi. Points of Inflection a. The graph of a polynomial function of degree n, with n 2 , has at most n 2 points of inflection. b. The graph of a polynomial function of odd degree, with n 2 , has at least one point of inflection. Complex Numbers i. Properties of the Complex Number System 1. The complex number system contains all real numbers. 2. Addition, subtraction, multiplication, and division of complex numbers obey the same rules of arithmetic that hold in the real number system, with one exception: a. The exponent laws hold for integer exponents but not necessarily for fractional ones. 3. The complex number system contains a number i such that i 2 1 . 4. Every complex number can be written in the standard form a bi , where a and b are real numbers. 5. a bi c di if and only if a c and b d. ii. Powers of i i1 i i 2 1 i 3 i 2 i i i4 i2 i2 1 i5 i 4 i i iii. Square Roots of negative numbers Let b be a positive real number. 1. b i b 2. Always write square roots of negative numbers in terms of i before doing any simplification. iv. Complex Solutions 1. Complex conjugate a. The conjugate of the complex number a bi is a bi . b. Every real number is its own conjugate. 2. Complex Solutions a. Complex solutions can be found for equations that have no real solutions. b. Every quadratic equation with real coefficients has solutions in the complex number system. c. If a bi is a solution of a polynomial with real coefficients, then its conjugate a bi is also a solution of the equation. The Fundamental Theorem of Algebra EVERY NONCONSTANT POLYNOMIAL HAS A ZERO IN THE COMPLEX NUMBER SYSTEM. i. Factorization over the Complex Numbers Let f x be a polynomial of degree n 0 with leading coefficient a. Then there are n, not necessarily distinct, complex numbers c1 , c2 ,...cn such that f x a x c1 x c2 ... x cn . Furthermore, c1 , c2 ,...cn are the only zeros of f. ii. Number of Zeros Every polynomial of degree n 0 has at most n different zeros in the complex number system. iii. Conjugate Zero Theorem Let f x be a polynomial with real coefficients. If the complex number z is a zero of f, then its conjugate z is also a zero of f. iv. Factorization over the Real Numbers Every nonconstant polynomial with real coefficients can be factored as a product of linear and irreducible quadratic polynomials with real coefficients in such a way that the quadratic factors, if any, have no real zeros.
