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Math Project

Math Project. Andy Frank Andrew Trealor Enrico Bruschi. Chapter 2.1. Types of polynomials Degree Name Example 0 Constant 5 1 Linear 3x+5 2 Quadratic x 2 +3x+5 3 Cubic x 3 + x 2 +3x+5

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Math Project

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  1. Math Project Andy Frank Andrew Trealor Enrico Bruschi

  2. Chapter 2.1 • Types of polynomials • Degree Name Example 0 Constant 5 1 Linear 3x+5 2 Quadratic x2+3x+5 3 Cubic x3+ x2+3x+5 4 Quartic -3x4+5 5 Quintic x5+3x4-3x3+11

  3. Practice on Polynomials • Give the degree and name of each of the following • 22x2+3 A. Degree 2 and Quadratic • 55 A. Degree 0 and Constant • 11x5+11x3+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-x2 A. p(x2) • 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) = anxn + an-1xn-1 + … + a0, where a0 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 a0, and q must be a factor of an.

  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 anxn + an-1xn-1 + … + a0= 0, with an does not equal 0: The sum of the roots is –(an-1-an) The product of the sum is (a0-an) if n is even - (a0-an) if n is odd

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