Today in Pre-Calculus

1
Today in Pre-Calculus

• Go over homework
• Notes Remainder
• and Factor Theorems
• Homework

2
Remainder Theorem

• If a polynomial f(x) is divided by x k, then
  the remainder is r f(k).
• - Find the remainder without doing synthetic
  division.
• Ex Use the remainder theorem to determine the
  remainder when f(x) x2 4x 5 is divided by
• x 3
• k 3 (3)2 4(3) 5 -8
• x 2
• k -2 (-2)2 4(-2) 5 7
• x 5
• k 5 (5)2 4(5) 5 0

3
Remainder Theorem

• Because the remainder in example c is zero, we
  know that x 5 divides evenly into f(x).
• Therefore, 5 is a zero or x intercept of the
  graph of f(x). And 5 is a solution or root of the
  equation f(x) 0.

4
Fundamental Connections for Polynomial Functions

• For a polynomial function f and a real number k,
  the following statements are equivalent
• 1. x k is a solution (or root) of the
  equation f(x) 0
• 2. k is an x-intercept of the graph of y f(x)
• k is a zero of the function f
• x k is a factor of f(x)

5
Factor Theorem

• A polynomial function f(x) has a factor x k iff
  f(k) 0.

6
Examples

• Use the factor theorem to determine if the first
  polynomial is a factor of the second polynomial.
• x 2 4x3 2x2 x 5
• k -2 4(-2)3 2(-2)2 (-2) 5
  -32 8 2 5 -47
• ?0, therefore, x 2 is not a factor of 4x3
  2x2 x 5
• 2. x 2 x3 2x2 5x 26
• k -2 (-2)3 2(-2)2 5(-2) 26 8
  8 10 26 0
• therefore, x 2 is a factor of x3 2x2 5x
  26

7
Writing Polynomial Functions

• Example Leading Coefficient 3
• Degree 3
• Zeros -4, 3, -1
• so factors are x 4, x 3, x 1
• 3(x 4)(x 3)(x 1)
• (3x 12)(x2 2x 3)
• 3x3 6x2 33x 36

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Writing Polynomial Functions

• Example Leading Coefficient 2
• Degree 3
• Zeros -3, -2, 5
• so factors are x 3, x 2, x 5
• 2(x 3)(x 2)(x 5)
• (2x 6)(x2 3x 10)
• 2x3 38x 60

9
Homework

• Pg. 223 13-24all, 27-30all