Rational Zero Theorem

1
Rational Zero Theorem

• Chapter 7.6

2
Vocabulary

• Rational Zero Theorem - theorem, that will will
  help choose some possible zeroes to test in a
  polynomial function.
• Corollary (Integral Zero Theorem) - states that
  if the coefficient of a polynomial function are
  integers such that a0 1 and an?0, any rational
  zeros of the function must be factors of an.

3
Steps for identifying possible rational zeroes

• Arrange the polynomials in descending order
• 2x3-11x2 912x 2x3-11x2 12x
  9
• Write down all the factors of the constant term.
  (These are all the possible values of p.)
• 1,
  3, 9
• Write down all the factors of the leading
  coefficient. These are all the possible values of
  q.
• 
  1, 2
• Write down all the possible values (factors) of p
  over q. Remember that since factors can be
  negative, p over q and - p over - q must
  both be included. Simplify each value and cross
  out any duplicates.
• 1, 3, 9,
  1/2, 3/2, 9/2

4
Examples

• List all possible rational
• zeroes of each function
• 1. f(x)3x 2x2 4 x5
• x5 2x2 3x 4
• p(1, 2, 4,) q(1)
• p/q(1, 2, 4, 1/1, 2/1, 4/1)
• 1, 2, 4

• 2. f(x) 9x35x2 2x3
• p(1, 3) q(1, 3, 9)
• p/q(1, 3, 1/3)
• Find The error
• 3. 25x4 15x3 16x2 4x 2
• p( 1, 5, 25,) q (1, 2)
• p/q 1, 1/5, 1/25, 2, 2/5,
  2/25

5
Steps for identifying all rational zeroes

• Step 1 Follow the same steps as you would if you
  were finding the possible rational zeroes
  of a function
• Step 2 Use synthetic division

6
Example

• X3-3x2-2
• 1, 2
• SEE BOARD

7
Steps for identifying all zeroes

• Step 1 Follow the same steps as you would if you
  were finding the possible rational zeroes
• Step 2 Use synthetic division to get
• Step 3 use synthetic division to find one factor
  of the quotient and continue this process until
  the polynomial has been completely factored

8
Example

• SEE BOARD