Slide 1: POLYNOMIAL FUNCTIONS A POLYNOMIAL is a monomial or a sum of monomials. A POLYNOMIAL IN ONE VARIABLE is a polynomial that contains only one variable. Example: 5x 2 + 3x - 7
Slide 2: POLYNOMIAL FUNCTIONS The DEGREE of a polynomial in one variable is the greatest exponent of its variable. A LEADING COEFFICIENT is the coefficient of the term with the highest degree. What is the degree and leading coefficient of 3x 5 – 3x + 2 ?
Slide 3: POLYNOMIAL FUNCTIONS A polynomial equation used to represent a function is called a POLYNOMIAL FUNCTION . Polynomial functions with a degree of 1 are called LINEAR POLYNOMIAL FUNCTIONS Polynomial functions with a degree of 2 are called QUADRATIC POLYNOMIAL FUNCTIONS Polynomial functions with a degree of 3 are called CUBIC POLYNOMIAL FUNCTIONS
Slide 4: POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find f(-2) if f(x) = 3x 2 – 2x – 6 f(-2) = 3(-2) 2 – 2(-2) – 6 f(-2) = 12 + 4 – 6 f(-2) = 10
Slide 5: POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find f(2a) if f(x) = 3x 2 – 2x – 6 f(2a) = 3(2a) 2 – 2(2a) – 6 f(2a) = 12a 2 – 4a – 6
Slide 6: POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find f(m + 2) if f(x) = 3x 2 – 2x – 6 f(m + 2) = 3(m + 2) 2 – 2(m + 2) – 6 f(m + 2) = 3(m 2 + 4m + 4) – 2(m + 2) – 6 f(m + 2) = 3m 2 + 12m + 12 – 2m – 4 – 6 f(m + 2) = 3m 2 + 10m + 2
Slide 7: POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find 2 g (-2a) if g (x) = 3x 2 – 2x – 6 2 g (-2a) = 2[3(-2a) 2 – 2(-2a) – 6 ] 2 g (-2a) = 2[12a 2 + 4a – 6] 2 g (-2a) = 24a 2 + 8a – 12
Slide 8: POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = 3 Constant Function Degree = 0 Max. Zeros: 0
Slide 9: POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1 Max. Zeros: 1
Slide 10: POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x 2 + 3x + 2 Quadratic Function Degree = 2 Max. Zeros: 2
Slide 11: POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x 3 + 4x 2 + 2 Cubic Function Degree = 3 Max. Zeros: 3
Slide 12: POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x 4 + 4x 3 – 2x – 1 Quartic Function Degree = 4 Max. Zeros: 4
Slide 13: POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x 5 + 4x 4 – 2x 3 – 4x 2 + x – 1 Quintic Function Degree = 5 Max. Zeros: 5
Slide 14: POLYNOMIAL FUNCTIONS END BEHAVIOR Degree: Even Leading Coefficient: + End Behavior: As x - ∞; f(x) +∞ As x + ∞; f(x) +∞ f(x) = x 2
Slide 15: POLYNOMIAL FUNCTIONS END BEHAVIOR Degree: Even Leading Coefficient: – End Behavior: As x - ∞; f(x) -∞ As x + ∞; f(x) -∞ f(x) = -x 2
Slide 16: POLYNOMIAL FUNCTIONS END BEHAVIOR Degree: Odd Leading Coefficient: + End Behavior: As x - ∞; f(x) -∞ As x + ∞; f(x) +∞ f(x) = x 3
Slide 17: POLYNOMIAL FUNCTIONS END BEHAVIOR Degree: Odd Leading Coefficient: – End Behavior: As x - ∞; f(x) +∞ As x + ∞; f(x) -∞ f(x) = -x 3
Slide 18: REMAINDER AND FACTOR THEOREMS f(x) = 2x 2 – 3x + 4 2 2 -3 4 2 4 1 2 6 Divide the polynomial by x – 2 Find f(2) f(2) = 2(2) 2 – 3(2) + 4 f(2) = 8 – 6 + 4 f(2) = 6
Slide 19: REMAINDER AND FACTOR THEOREMS f(x) = 3x 5 – 4x 3 + 5x - 3 Find f(-3) Try this one: Remember – Some terms are missing When synthetic division is used to evaluate a function, it is called SYNTHETIC SUBSTITUTION .
Slide 20: REMAINDER AND FACTOR THEOREMS The binomial x – a is a factor of the polynomial f(x) if and only if f(a) = 0. FACTOR THEOREM
Slide 21: REMAINDER AND FACTOR THEOREMS Is x – 2 a factor of x 3 – 3x 2 – 4x + 12 2 1 -3 -4 12 1 2 -1 -2 -6 -12 0 Yes, it is a factor, since f(2) = 0. Can you find the two remaining factors?
Slide 22: REMAINDER AND FACTOR THEOREMS (x + 3)( ? )( ? ) = x 3 – x 2 – 17x - 15 Find the two unknown ( ? ) quantities.