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Polynomials and Polynomial Functions Section 5.3

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

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Terms Number (example: 1, 0, -2, 121) Variable (example: x, y, z) Product of numbers and/or variables (example: 3a 2 b 4, 2y, 5x 2 ) Quotient of numbers and/or variables (example: )

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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, 23x 2 y, ⅓a 5 –Note: Terms like 1/t or x -2 are not monomials

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Terms Number (example: 1, 0, -2, 12, ⅓) Variable (example: x, y, z) Product of numbers and/or variables (example: 3a 2 b 4, 2y, 5x 2 ) Quotient of numbers and/or variables (example: ) Monomial or Not? Yes No

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Types of Polynomials Polynomial –A monomial or a sum of monomials –Example: 4x + 7, ⅓t 2, 6a + 7, 6, 0 –When polynomial is sum of monomials, each monomial is called a term of the polynomial

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Types of Polynomials ID the terms of the polynomial 3t 4 – 5t 6 – 4t - 2

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

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Degrees and Coefficients Degree of a term –The number of variable factors in that term –The degree of 7t 2 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

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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: 3x 2 – 8x 3 + 5x 4 + 7x - 6

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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

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Polynomial Functions Polynomial function – function involving a polynomial expression P(x) = 5x 7 + 3x 5 – 4x 2 -5 Linear function – degree of polynomial is 1 f(x) = 4x + 5 Quadratic function – degree is 2 f(x) = 3x 2 – 4x + 5 Cubic function – degree is 3 f(x) = 2x 3 + 3x 2 – 4x + 5 Quartic function – degree is 4 f(x) = x 4 – 2x 3 + 3x 2 – 4x + 5

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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

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Next up: Addition And Subtraction of Polynomials Read Section 5.4

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