4.1: Polynomial Functions

1. 4.1: Polynomial Functions Objectives: Define a polynomial Divide Polynomials Apply the remainder theorem, the factor theorem, and the connections between remainders and factors Determine the maximum number of zeros of a polynomial Michigan Standards: P4.3

2. Defn. of a Polynomial Function • A polynomial function is a function whose rule is given by a polynomial • where are real numbers with and n is a nonnegative number.

3. Defn. of a Polynomial Function • An is called a coefficient. The number in front of a variable. • A0 is called the constant term, there are no variables with the number. • Characteristics of a polynomial • All exponents are whole numbers • No variable is contained in a denominator • No variable is under a radical

4. A polynomial that consists of only a constant term is called a constant polynomial. • The zero polynomial is the constant polynomial 0. • The exponent of the highest power of x that appears with nonzero coefficient is the degree of the polynomial. • The nonzero coefficient of the highest power of the variable is the leading coefficient.

5. First-degree polynomials are called linear functions. • Second-degree polynomial functions are called quadratic functions. • Third-degree polynomial functions are called cubic functions. • Fourth-degree polynomial functions are called quartic functions.

6. Long Division of Polynomials • Divide 3x4 – 8x2 – 11x + 1 by x-2 (Hint: don’t forget about the x3) Subtract when dividing.

7. Synthetic Division • -c Coefficients • Set up the following: • Divide 3x4 – 8x2 – 11x + 1 by x-2

8. Synthetic Division of Polynomials • Divide 3x4 – 8x2 – 11x + 1 by x-2 (Hint: don’t forget about the x3) Add when doing synthetic division.

9. Assignment Part 1. • Page 248 • Questions 1-16 all. Follow all the directions, and SHOW ALL YOUR WORK!

10. The Division Algorithm • If a polynomial f(x) is divided by a nonzero polynomial h(x) then there is a quotient polynomial q(x) and a remainder polynomial r(x) such that where r(x)=0 or r(x) has degree less than the degree of the divisor, h(x). Dividend Divisor Remainder Quotient

11. Remainders and Factors • If the remainder is 0, the divisor and the quotient are factors of the dividend. • Remainder Theorem • If a polynomial f(x) is divided by x - c, then the remainder is f(c). • Factor Theorem • A polynomial function f(c) has a linear factor x–a iff f(a) = 0.

12. Determine if 2x2+1 is a factor of 6x3-4x2+3x-2

13. Find the Remainder when x79+3x24+5 is divided by x-1

14. Find the Remainder when 3x4-8x2+11x+1 is divided by x+2

15. Show x-3 is a factor of x3-4x2+2x+3. If so, Write in factored form

16. Assignment Part 2. • Page 248 and 249 • Questions • 18-26 Evens • 32-40 Evens • 42-46 Evens • Read and follow all directions!

17. Zeros, x-intercepts, solutions, and factors Let f(x) be a polynomial. If c is a real number that satisfies any of the following statements, then c satisfies all the statements. • c is a zero of the function f • c is an x-intercept of the graph of f • x = c is a solution, or root, of the equation f(x) = 0 • x – c is a factor of f

18. f(x)= 15x3-x2-114x+72 • Find • a. The x-intercepts • b. The Zeros • c. The solutions to f(x) • d. The linear factors

19. Find a polynomial with zeros of 1, 2, 3, and -5

20. Number of Zeros • A polynomial of degree n has at most n distinct real zeros.

21. What are the maximum number of Zeros in the following polynomial • 18x4 – 51x3 – 187x2 – 56x + 80

22. Assignment Part 3 • P. 249-250 • 28-30 Evens • 56-58 Evens