Warm-up

1
Warm-up

• Given the following polynomials and one factor,
  find all the real zeros of the polynomials.
• f(x) 2x3 2x2 8x 8 (x1) is a factor
• f(x) 2x3 3x2 14x 15 (x1) a factor

2
Pumpkin Project

• Double Quiz Grade
• Due October 26
• Design a Pumpkin Face using our 8 Parent
  Functions.
• Use all functions at least ONE TIME
• No More than 10 equations, no less than 8
• MUST DO BOTH Calculator part and Graphing Part to
  be graded!
• TURN IN Early for 10 points BONUS

3
Zeros of Polynomials

• 2.5

4
Solving for solutions from factors

• Set the factor equal to zero
• Solve for x
• Solutions can be real or imaginary

5
Rational Zero Theorem

• Given f(x) anxn a1x a0 has integer
  coefficients, then every rational zero has the
  format

6
EXAMPLE 1
List possible rational zeros
List the possible rational zeros of f using the
rational zero theorem.
a. f (x) x3 2x2 11x 12
Factors of the constant term 1, 2, 3, 4,
6, 12
Factors of the leading coefficient 1
Simplified list of possible zeros 1, 2, 3,
4, 6, 12
7
List possible rational zeros
EXAMPLE 1
b. f (x) 4x4 x3 3x2 9x 10
Factors of the constant term 1, 2, 5, 10
Factors of the leading coefficient 1, 2, 4
8
B. List all of the possible rational zeros of
f(x) x3 3x 24.
9
Properties of Polynomial Equations

• A polynomial of degree n, has n roots (counting
  multiple roots separately)
• If a bi is a root, then a bi is also a root.
  Complex roots occur in conjugate pairs!

10
Complex Solutions
If this is 1 solution
Then this is another solution
11
Determining Types of Solutions

• Degree tells total number of complex solutions
• Complex solutions include both real and imaginary
  solutions
• Imaginary roots occur in conjugate pairs
• Lets look at some possibilities on the next slide

12
Solutions based on degree

• f(x) 3x4 2x2 7x 4
• 4 complex solutions
• Possible combinations
• 4 real, 0 imaginary
• 2 real, 2 imaginary
• 0 real, 4 imaginary

13
Solutions based on degree

• f(x) 5x5 4x4 2x2 - 7
• 5 complex solutions
• Possible combinations
• 5 real, 0 imaginary
• 3 real, 2 imaginary
• 1 real, 4 imaginary

14
BIG Picture

• The degree of the polynomial tells us how many
  complex solutions we have and the most rational
  zeros we can have
• The Rational Zero Theorem tells us POSSIBLE
  rational zeros
• WE need to use other methods to find the actual
  zeros (solutions)
• Mondays Topic!!

15
Homework

• WS 4-3