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
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
28-30 Evens
56-58 Evens