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Polynomials and Polynomial Functions

Polynomials and Polynomial Functions. Section 5.3. Overview. Terms Types of Polynomials Degree and Coefficients Combining Like Terms Polynomial Functions Graphs of Polynomial Functions. Terms. Number (example: 1, 0, -2, 121) Variable (example: x, y, z)

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Polynomials and Polynomial Functions

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  1. Polynomials and Polynomial Functions Section 5.3

  2. Overview • Terms • Types of Polynomials • Degree and Coefficients • Combining Like Terms • Polynomial Functions • Graphs of Polynomial Functions

  3. Terms • Number (example: 1, 0, -2, 121) • Variable (example: x, y, z) • Product of numbers and/or variables (example: 3a2b4, 2y, 5x2) • Quotient of numbers and/or variables (example: )

  4. Types of Polynomials • Monomial • Product of constants or variables • Variables only raised to whole number exponents (i.e. 0 or positive integer) • Example: 7, t, 23x2y, ⅓a5 • Note: Terms like 1/t or x-2 are not monomials

  5. Number (example: 1, 0, -2, 12, ⅓) Variable (example: x, y, z) Product of numbers and/or variables (example: 3a2b4, 2y, 5x2) Quotient of numbers and/or variables (example: ) Monomial or Not? Yes Yes Yes No Terms

  6. Types of Polynomials • Polynomial • A monomial or a sum of monomials • Example: 4x + 7, ⅓t2, 6a + 7, 6, 0 • When polynomial is sum of monomials, each monomial is called a term of the polynomial

  7. Types of Polynomials • ID the terms of the polynomial 3t4 – 5t6 – 4t - 2

  8. Types of Polynomials • Polynomial with one term • Monomial • 4x2 • Two terms • Binomial • 2x + 4 • Three terms • Trinomial • 3t2 + 4t + 7 • Four terms • No special name for polynomials with four or more terms • 4x3 - 5x2 + xy - 8

  9. Degrees and Coefficients • Degree of a term • The number of variable factors in that term • The degree of 7t2 is 2 (t and t) • Coefficient • The part of the term that is a constant factor (i.e. the numeral) • The coefficient of 3x is 3

  10. Degree and Coefficients • Leading term – term of highest degree • Leading coefficient – coefficient of the leading term • Degree of the polynomial – degree of the leading term • Example: 3x2 – 8x3 + 5x4 + 7x - 6

  11. Combining Like terms • Like terms (or similar terms) • Constant terms • Terms containing the same variable(s) raised to the same power(s) • To simplify certain polynomials, you can often combine, or collect, like terms • Adding or subtracting like terms • Write solution in descending order with term of highest degree first, followed by term of next highest degree, and so on

  12. Polynomial Functions • Polynomial function – function involving a polynomial expression P(x) = 5x7 + 3x5 – 4x2 -5 • Linear function – degree of polynomial is 1 f(x) = 4x + 5 • Quadratic function – degree is 2 f(x) = 3x2 – 4x + 5 • Cubic function – degree is 3 f(x) = 2x3 + 3x2 – 4x + 5 • Quartic function – degree is 4 f(x) = x4 – 2x3 + 3x2 – 4x + 5

  13. Graphs of Polynomial Functions • Common characteristics (refer to graphs on p. 379) • Smooth line • Continuous line • Domain is all real numbers, unless otherwise specified • Range

  14. Next up:Addition And Subtraction of Polynomials Read Section 5.4

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