MA112
Section 106
Dr. Byrne
Fall 2009
Worksheet: Polynomial Division (4.3)
Find the quotient Q (x) and the remainder R (x ) when the polynomial P (x ) is divided by the
polynomial D(x) .
1. P( x)  2 x 2  3x  3 ;
D( x)  x  2
2. P( x)  3x 2  3x  2 ;
D( x)  x  1
3. P( x)  x3  x 2  2 ;
D( x)  x  1
4. P( x)  3x 4  2 x3  x  2 ;
D( x)  x 2  2 x  1
Worksheet: Factor Theorem (4.3)
Factor Theorem: x  c is a factor of P (x ) if P (c)  0.
Show that x  c is a factor of P (x ) for the given value of c, and (b) factor P (x ) completely.
P( x)  x 3  5 x 2  8 x  4 ; c  1
(b) factor P (x ) completely
(a) P (c ) 
Worksheet: Rational ZerosTheorem (4.4)
The Rational Zeros Theorem
Suppose that P( x)  a n x n  a n 1 x n 1    a 2 x 2  a1 x  a0 .
Then any rational root of P(x) will be p / q where p divides the constant term a0 and q divides the
leading coefficient an.
Determine all possibilities for rational zeros.
x 4  2 x 3  x 2  3x  4  0
6 x 5  24 x 4  40 x 3  48 x 2  24 x  4  0
Find all rational zeros of the polynomial by (a) finding all possible rational zeros and (b) graphing
the polynomial to determine which ones are actually zeros. Once you have found the zeros, (c)
factor the polynomial completely.
21. x 3  3 x 2  4
(a) Possible zeros:
(b) sketch
x
y
(c) Factor the polynomial completely:
23. 2 x 4  x3  5 x 2  2 x
(a) Possible zeros:
(c) Factor the polynomial completely:
(b) sketch
x
y