5.2 WARM-UP

Evaluate and Graph Polynomial Functions 5.2 Evaluate and Graph Polynomial Functions A polynomial is a monomial or a sum of monomials A polynomial function has exponents that are all whole numbers and the coefficients are all real numbers. The leading coefficient is the coefficient in front of the highest exponent The degree of the polynomial is the highest power To write a polynomial in standard form write the exponents in descending order Example Degree Type f(x) = -14 Constant f(x) = 5x - 7 1 Linear f(x) = 2x2 + x - 9 2 Quadratic f(x) = x3 – x2 + 3x 3 Cubic f(x) = x4 + 2x - 1 Quartic 4

1 a. h (x) = x4 – x2 + 3 4 b. g (x) = 7x – 3 + πx2 Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. Already in standard form 4 a. h (x) = x4 – x2 + 3 1 Degree – 4 (quartic) Leading coefficient = 1 g(x) = πx2 + 7x – 3 standard form b. g (x) = 7x – 3 + πx2 degree – 2 (quadratic) leading coefficient = π The function is not a polynomial function because the term 3x – 1 has an exponent that is not a whole number. c. f (x) = 5x2 + 3x -1 – x The function is not a polynomial function because the term 2x does not have a variable base and an exponent that is a whole number. d. k (x) = x + 2x – 0.6x5 Already in standard form Degree: 1 (linear) Leading coefficient of – 2. e. f (x) = – 2x + 13

f (x) = 2x4 – 5x3 – 4x + 8 when x = 3. f (x) = 2x4 – 5x3 – 4x + 8 Use direct substitution to evaluate f (x) = 2x4 – 5x3 – 4x + 8 when x = 3. f (x) = 2x4 – 5x3 – 4x + 8 f (3) = 2(3)4 – 5(3)3 – 4(3) + 8 g (x) = x 3 – 5x2 + 6x + 1; x = 4 = 162 – 135 – 12 + 8 g (x) = 4 3 – 5(4)2 + 6(4) + 1 = 23 = 64 – 80 + 24 + 1 = 9 f (x) = x4 + 2x3 + 3x2 – 7; x = – 2 f (–2) = (–2) 4 + 2 (–2) 3 + 3 (–2 )2 – 7 = 16 – 16 + 12 – 7 = 5

Divide using Synthetic Substitution (synthetic division) (4x3 – 9x2 – 10x – 2) (x - 3) Change sign of divisor and use only coefficients of Dividend 3 4 -9 -10 -2 12 9 -3 4 3 -1 -5 The last number that you get is the remainder of the equation and also your answer to the problem. Try (x4 – 6x3 – 2x – 10) (x + 1) The exponents need to be in descending order. If there is a missing exponent, you need to write the coefficient as “0”. -1 1 -6 0 -2 -10 -1 7 -7 9 _________________ 1 -7 7 -9 -1

Use Synthetic Substitution x4 + x3 – 13x2 – 25x -12; x = 0 and x = -4 (don’t change the sign of the divisor when it is x = #) 1 1 -13 -25 -12 _____________________ 1 1 -13 -25 -12 By Synthetic Substitution…f(0) = -12 ( you could get the same answer by direct substitution – but that’s not what we’re asking you to do) -4 1 1 -13 -25 -12 -4 12 4 84 _____________________ 1 -3 -1 -21 72 HOMEWORK 5.2 p. 341 #3-23 f(4) = 72