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Today in Pre-Calculus
• Go over homework
• Notes: Remainder
and Factor Theorems
• Homework
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:
a)x – 3
k=3
(3)2 – 4(3) – 5 = -8
b)x + 2
k = -2
(-2)2 – 4(-2) – 5 = 7
c)x – 5
k=5
(5)2 – 4(5) – 5 = 0
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.
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)
3. k is a zero of the function f
4. x – k is a factor of f(x)
Factor Theorem
A polynomial function f(x) has a factor x – k iff f(k) = 0.
Examples
Use the factor theorem to determine if the first polynomial is
a factor of the second polynomial.
1. 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
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
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
Homework
• Pg. 223: 13-24all, 27-30all