1. Math Project Andy Frank Andrew Trealor Enrico Bruschi

2. Chapter 2.1 Types of polynomials DegreeNameExample 0 Constant 5 1Linear 3x+5 2 Quadratic x 2 +3x+5 3 Cubic x 3 + x 2 +3x+5 4 Quartic-3x 4 +5 5 Quinticx 5 +3x 4 -3x 3 +11

3. Practice on Polynomials Give the degree and name of each of the following 1. 22x 2 +3 A. Degree 2 and Quadratic 2. 55 A. Degree 0 and Constant 3. 11x 5 +11x 3 +11 A. Degree 5 and Quintic

4. Chapter 2.2 The Remainder Theorem When a polynomial P(x) is divided by x – a, the remainder is P(a). The Factor Theorem For a polynomial P(x), x – a is a factor iff P(a) = 0

5. Practice on Remainders Give the remainders of the following: When p(5) is divided by 5-x 2 A. p(x 2 ) When t(8421) is divided by 8421-21 A. t(21) When c(x) is divided by x-b A. c(b)

6. Chapter 2.5 The Location Principle If P(x) is a polynomial with real coefficieants and a and b are real numbers such that P(a) and P(b) have opposite signs, then between a and b there is at least one real root r of the equation P(x) = 0.

7. Chapter 2.6 The Rational Root Theorem Let P(x) be a polynomial of degree n with integral coefficients and a nonzero constant term: P(x) = a n x n + a n-1 x n-1 + … + a 0, where a 0 does not equal 0 If one of the roots of the equation P(x) = 0 is x = p/q where p and q are nonzero intergers with no common factor other than 1, then p must be a factor of a 0, and q must be a factor of a n.

8. Chapter 2.7 Theorem 1 (Fundamental Theorem of Algebra) In the complex number system consisting of all real and imaginary numbers, if P(x) is a polynomial of n (n>0) with complex coefficients, then the equation P(x) = 0 has exactly n roots (provided a double root is counted as 2 roots, a triple root is counted as 3 roots, and so on.)

9. Chapter 2.7 (cont.) Theorem 2 (Complex Conjugates Theorem) If P(x) is a polynomial with real coefficients, and a + bi is an imaginary root of the equation P(x) = 0, then a – bi is also a root.

10. Chapter 2.7 (cont.) Theorem 3 Suppose P(x) is a polynomial with rational coefficients, and a and b are rational numbers, such that the square root of b is irrational. If a + the square root of b is a root of the equation P(x) = 0, then a – the square root of b is also a root.

11. Chapter 2.7 (cont.) If P(x) is a polynomial of oddd degree with real coefficients, then the equation P(x) = 0 has at least on real root.

12. Chapter 2.7 (cont.) Theorem 5 For the equation a n x n + a n-1 x n-1 + … + a 0 = 0, with a n does not equal 0: The sum of the roots is –(a n-1 -a n ) The product of the sum is (a 0 -a n ) if n is even - (a 0 -a n ) if n is odd