2 2 B Higher Degree Polynomial Functions Relative

2. 2 B Higher Degree Polynomial Functions • Relative Extrema: Turns (humps) of graph § AT MOST = degree – 1 § Relative maximum = MAXIMA § Relative minimum = MINIMA

Real Zeros of Polynomial Functions • Zeros = x-intercepts = real solutions • • • N = total # of solutions X = a = zero of function f X = a = Solution of polynomial f(x)=0 (x – a) = factor of polynomial f(x) (a, 0) = x – intercept of graph of f

Examples • 1. f(x) = x³ + 3 x² - 10 x N= Total number of solutions = Factors = ( )( ) Zeros = solutions = X-intercepts = ( , 0)

More examples • 2. Given the real zeros (solutions) x = 0, 0, -1, 3 Find: A. ) x-intercepts = ( , 0) ( B) Factors = C) Polynomial f(x) = D) Degree, n = , 0) ( , 0 )

Multiplicity: Repeated Answers •

Finding the real zeros • Algebraically – Factor – If a factor has x², factor more or use quadratic formula • Graphically – Put into calculator in y= – Use zero feature (2 nd , trace)

Examples: • 3. Find all the real zeros and determine the multiplicity of each for f(x) = x³ - 4 x² + 4 x • 4. Use the calculator to graph # 3 & find any real zeros and extrema (round to 3 places)

More examples • 5. Find the zeros algebraically & state multiplicity of each, then use calculator to graph and find zeros. • A) f(x) = 3 x² - 12 x + 3 • B) g(x) = ½ x ³ (x² - 16)(x – 4)