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2.1 Evaluate and Graph Polynomial Functions

2.1 Evaluate and Graph Polynomial Functions. Objectives: Identify, evaluate, add, and subtract polynomials Classify polynomials, and describe the shapes of their graphs. What is a Polynomial?. 1 or more terms Exponents are whole numbers (not a radical)

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2.1 Evaluate and Graph Polynomial Functions

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  1. 2.1 Evaluate and Graph Polynomial Functions Objectives: Identify, evaluate, add, and subtract polynomials Classify polynomials, and describe the shapes of their graphs

  2. What is a Polynomial? • 1 or more terms • Exponents are whole numbers (not a radical) • Coefficients are all real numbers (no imaginary #’s) • It is in Standard Form when the exponents are written in descending order.

  3. Definitions for Polynomials NOT QUOTIENT! i.e. “x” can’t be on bottom!!!! • Monomial: a numeral, variable, or the product of a numeral and one or more variables Ex: • Constant: a monomial w/ no variables Ex: • Coefficient: numerical factor in a monomial Ex: • Degree of a Monomial: sum of exponents of its variables Ex: See Below Give the degree for the following monomial. 4x2y3z _________ ab4c2 ________ 8 ________ 7 6 0

  4. Definitions for Polynomials • Polynomial: is many (more than 1) monomials connected by addition or subtraction. (5x + 4) (2x2 + 3x – 2) Binomial - ___________ Trinomial - ______________ • Degree of the Polynomial: is the degree of it’s highest monomial term Example: Give the degree of the polynomial. 4x3 + 6x2 -8x5 – 6 ________

  5. Classification of a Polynomial n = 0 constant 3 linear n = 1 5x + 4 quadratic n = 2 2x2 + 3x - 2 cubic n = 3 5x3 + 3x2 – x + 9 quartic 3x4 – 2x3 + 8x2 – 6x + 5 n = 4 n = 5 -2x5 + 3x4 – x3 + 3x2 – 2x + 6 quintic

  6. Classify each polynomial by degree and by number of terms. a) 5x + 2x3 – 2x2 b) x5 – 4x3 – x5 + 3x2 + 4x3 c) x2 + 4 – 8x – 2x3 d) 3x3 + 2x – x3 – 6x5 e) 2x + 5x7 quintic trinomial cubic polynomial Not a polynomial cubic trinomial quadratic monomial 7th degree binomial

  7. 1 a. h (x) = x4 – x2 + 3 EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient. 4 SOLUTION a. The function is a polynomial function that is already written in standard form. It has degree 4 (quartic) and a leading coefficient of 1.

  8. b. The function is a polynomial function written as g(x) = πx2 + 7x – 3 in standard form. It has degree 2(quadratic) and a leading coefficient of π. b. g (x) = 7x – 3 + πx2 EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient. SOLUTION

  9. EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient. c. f (x) = 5x2 + 3x –1– x SOLUTION c. The function is not a polynomial function because the term 3x – 1 has an exponent that is not a whole number.

  10. EXAMPLE 1 Identify polynomial functions Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient. d. k (x) = x + 2x– 0.6x5 SOLUTION d. The function is not a polynomial function because the term 2xdoes not have a variable base and an exponent that is a whole number.

  11. for Examples 1 and 2 GUIDED PRACTICE Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. 2. p (x) = 9x4 – 5x – 2 + 4 1. f (x) = 13 – 2x not a polynomial function polynomial function; f (x) = –2x + 13;degree 1, type: linear, leading coefficient: –2 3. h (x) = 6x2 + π – 3x polynomial function; h(x) = 6x2 – 3x + π;degree 2, type: quadratic, leading coefficient: 6

  12. EXAMPLE 2 Evaluate by direct substitution Use direct substitution to evaluatef (x) = 2x4 – 5x3 –4x + 8whenx = 3. f (x) = 2x4 –5x3 – 4x+ 8 Write original function. f (3) = 2(3)4 – 5(3)3 – 4(3) + 8 Substitute 3 for x. = 162 – 135 – 12 + 8 Evaluate powers and multiply. = 23 Simplify

  13. for Examples 1 and 2 GUIDED PRACTICE Use direct substitution to evaluate the polynomial function for the given value of x. 4. f (x) = x4 + 2x3 + 3x2 – 7; x = –2 ANSWER 5 5. g(x) = x3– 5x2 + 6x + 1; x = 4 ANSWER 9

  14. Solving by Synthetic Substitution (Division) (x - 2) is a Factor of use x = 2 Use the Polynomials coefficients Drop 1st coefficient down Multiply Answer Remainder if there is any Add Down The Solution starts with one degree less than original

  15. Homework Pg. 69 1 – 20

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