6.5 Theorems About Roots of Polynomial Equations
Worksheet 1-8, Pg 333 13-24 all
1. The Remainder Theorem
If f (x) is a polynomial equation and if f (x) is divided by x  c , then the remainder is f (c ) .
Ex: f ( x)  x3  3x 2  x  3 ; x  2
f (___)  (____)3  3(____) 2  (____)  3
2. The Factor Theorem
Let f (x) be a polynomial equation.
A. If f (c ) = 0, then x  c ,is a factor of f (x) .
B. If x  c ,is a factor of f (x) , then f (c ) = 0.
Ex: f ( x)  x3  3x 2  x  3 ; x  1
So, you can determine factors ______ ways: ________________________ or _________________
We are going to ______________________________.
3. Irrational Root Theorem
Let a and b be rational numbers and let b be an irrational number. If a  b is a root of a
polynomial equation with rational coefficients, then the conjugate a  b also is a root.
Ex:
2 & ______
1 3
& ______
4. Imaginary Root Theorem
If a  bi is a root of a polynomial equation with rational coefficients, then the conjugate a  bi
also is a root.
1 i 3
Ex: 2  i 2 & ________
& _______
i 2 & _______
4
Use the Factor Theorem to determine whether x  c is a factor of the polynomial equation.
Ex 1 f ( x)  8x 3  3x 2  x  4 ; x - 1
Ex 2 f ( x)  x 4  x 2  2 x  2 ; c = -1
Ex 3 A polynomial equation has roots 2  9i and  7i . Find the additional roots.
Writing a polynomial equation from the roots.
Ex 4 Find a 3rd degree polynomial equation with roots -1, and 2-i.
HW Worksheet 6.5
Use the Factor Theorem to determine whether x  c is a factor of the polynomial equation.
1.
x 3  x 2  2 x  2  0; x  3
2.
x 3  x 2  2 x  2  0; x  1
3.
x 3  x 2  4 x  4  0; x  1
4.
x 3  x 2  4 x  4  0; x  2
5.
2 x 3  9 x 2  11x  8  0; x  2
6.
x 3  2 x 2  8 x  16  0; x  2
7.
x 3  2 x 2  5x  10  0; x  5
8.
x 4  2 x 2  15  0; x  5