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Published byShona Beatrice Gaines Modified over 8 years ago
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Polynomial Degree and Finite Differences Objective: To define polynomials expressions and perform polynomial operations
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Definitions: Polynomial Expression: The sum of terms containing the same variable raised to different powers Polynomial Function: A function where all exponents are whole numbers and the coefficients are all real numbers Written in the form a n x n + a n-1 x n-1 + … + a 1 x 1 + a 0
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Examples: Are each of the following polynomial functions? 1. f(x) = -5x 5 + 3x 4 – 2x + 8Yes 2. f(x) = 5 – 2xYes 3. f(x) = -8 Yes 4. f(x) = x 2 + 2 x No 5. f(x) = 6x 2 – 5x -1 No 6. No
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Definitions: Terms – the number of monomials that make up the polynomial Constant – the number without a variable Polynomials are named by their terms: Monomial: a polynomial that has one term Binomial: a polynomial that has two terms Trinomial: a polynomial that has three terms Polynomial: if there are more than three terms
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Definitions: Degree of a monomial – the sum of the exponents of the variables Degree of a polynomial – the degree of the term that has the highest degree Leading coefficient – the coefficient of the first term when the polynomial is in standard form Standard form – the degrees of the terms are written in descending order from left to right Even or Odd function – determined by the degree of the function
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Example: f(x) = 2 – x + 5x 4 – 3x 2 + 2x 3 Standard form: f(x) = 5x 4 + 2x 3 – 3x 2 – x + 2 Leading coefficient: 5555 Even or odd: Even Degree of each monomial: 4, 3, 2, 1, 0 Degree of the polynomial: 4444
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Practice: Identify the degree of each polynomial: 3x 4 – 2x 3 + 3x 2 – x + 7 4444 x 5 – 1 5555 0.2x – 1.5x 2 + 3.2x 3 3333 250 – 16x 2 + 20x 2222 x + x 2 – x 3 + x 4 – x 5 5555 5x 2 – 6x 5 + 2x 6 – 3x 4 + 8 6666
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Practice: Determine which of the expressions are polynomials. For each polynomial, state its degree and write it in standard form. If it is not a polynomial, explain why not. 1 + x 2 – x 3 polynomial?yesno Standard form:-x 3 + x 2 + 1 0.2x 3 + 0.5x 4 + 0.6x 2 polynomial:yesno Standard form:0.5x 4 + 0.2x 3 + 0.6x 2
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Practice: polynomial?yesno can’t have a variable in a denominator 25polynomial?yesno Standard form:25 polynomial?yesno Standard form: polynomial?yes no Can’t have a variable inside a radical
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Polynomial Operations To simplify polynomials: Combine like terms by adding or subtracting their coefficients To add polynomials: Combine like terms by adding or subtracting their coefficients To subtract polynomials: Remember that subtracting is the same as adding the opposite, so change the sign on the coefficient of each term being subtracted To multiply polynomials: Use the distributive property then combine like terms
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Practice Problems: (-x 2 + 2x – 1) + (6x 2 – x + 5) 5x 2 + x + 4 (2x 2 + 4x + 3) + (3x 2 – 9) 5x 2 + 4x – 6 (3x 2 – 2x + 6) + (-x 2 – 3) 2x 2 – 2x + 3 (7x 2 + x – 8) – (6x 2 – 3x + 5) 7x 2 + x – 8 – 6x 2 + 3x – 5 x 2 + 4x - 13
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Practice Problems (5x 3 – 5x 2 + 2x – 3) – (-x 2 + 4x + 3) 5x 3 – 5x 2 + 2x – 3 + x 2 – 4x – 3 5x 3 – 4x 2 – 2x – 6 x(7x 2 – 2x + 1) 7x 3 – 2x 2 + x (x 2 + 3)(x 2 + 2x – 4) x 4 + 2x 3 – 4x 2 + 3x 2 + 6x – 12 x 4 + 2x 3 –x 2 + 6x - 12
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Practice Problems (x + 3)(x – 1)(x + 5) (x 2 – x + 3x – 3)(x + 5) (x 2 + 2x – 3)(x + 5) x 3 + 5x 2 + 2x 2 + 10x – 3x – 15 x 3 + 7x 2 + 7x - 15
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