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Understanding Polynomials: Types, Theorems, and Practice Exercises

This project covers essential concepts of polynomials, including their types, degrees, and examples. It explains the Remainder Theorem, Factor Theorem, and provides insights into the Location Principle and Rational Root Theorem. Moreover, the Fundamental Theorem of Algebra and additional theorems regarding roots of polynomial equations are elaborated upon. Engaging practice exercises are included to reinforce understanding of polynomial properties and theorems. Ideal for students seeking to improve their grasp of polynomial mathematics.

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Understanding Polynomials: Types, Theorems, and Practice Exercises

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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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