5.5: Polynomial Long Division and Synthetic Division
HW: 5.4: p.357 (30-40 even, 46-54 even)

5 724

• Polynomial Long Division can be used for any two polynomials.
• Synthetic Division: the divisor has to be in the form x – k, where k is any constant.

Divide using polynomial long division:
2 x

5 6 x
2 
11 x

26


2 x
2 
7 x

10


 x

5


Divide:

2 x
3 
11 x
2 
13 x

44


 x

5


Divide:
 x
3 
4 x

6


 x

3


Divide:

4 x
4 
5 x

4
  x
2 
3 x

2


5.5: Given polynomial f(x) and a factor of f(x), factor f(x) completely.
f ( x )
 x
3 
10 x
2 
19 x

30 ; x

6

5.5: FINISH: Polynomial Long Division and Synthetic Division
Hw: 5.5: p.366 (8, 22, 28, 30, 32, 36)

Given polynomial f(x) and a factor of f(x), factor f(x) completely.
• Steps
1.) Divide the polynomial and the factor.
2.) Factor the answer.
3.) Write out all factors.

Given polynomial f(x) and a factor of f(x), factor f(x) completely.
f ( x )

2 x
3 
15 x
2 
34 x

21 ; x

1

Given polynomial f(x) and a zero of f(x), find the other zeros.
• Zeros: answers to the polynomial equation f(x) = 0.
• Process.
1.) Use the zero to factor the polynomial completely.
2.) Solve to find the other zeros.

Given polynomial f(x) and a zero of f(x), find the other zeros.
f ( x )
 x
3 
2 x
2 
21 x

18 ;

3

Given polynomial f(x) and a zero of f(x), find the other zeros.
f ( x )

10 x
3 
89 x
2 
12 x

27 ; 9

Given polynomial f(x) and a factor of f(x), factor f(x) completely.
f ( x )

3 x
3 
2 x
2 
61 x

20 ; x

5

Given polynomial f(x) and a zero of f(x), find the other zeros.
f ( x )

3 x
3 
34 x
2 
72 x

64 ;

4

5.6: Find Rational Zeros
HW tonight: p.374 (4-10 even)
Tomorrow: p.374 (14-20 even)
Next day: p.374-375 (24-30 even)

List all possible rational zeros using the rational zero theorem.
• Every rational zero of a function has the following form: p q
 factor factor of of constant t leading erm coefficien a
0 t a n

List all possible rational zeros using the rational zero theorem.
• Example: List the possible rational zeros for the function: f ( x )
 x
3 
2 x
2 
11 x

12

1 ,

2 ,

3 ,

4 ,

6 ,

12
Factors of the leading coefficient: 
1
Possible rational zeros: 
1
1
,

2
1
,

3
,

1
4
,

1
6
1
,

12
1
Possible rational zeros: 
1 ,

2 ,

3 ,

4 ,

6 ,

12

List all possible rational zeros using the rational zero theorem.
f ( x )

4 x
4  x
3 
3 x
2 
9 x

10

List all possible rational zeros using the rational zero theorem.
f ( x )

2 x
3 
3 x
2 
11 x

6

Find the zeros of a polynomial function.
• List the possible rational zeros of the function.
• Test the zeros using division. (Since the zeros are x-intercepts, when you divide you should end up with a remainder of zero.)
– Graph the function in the calculator to narrow your list. Only check reasonable values from the list.
– The number of zeros is the same as the degree of the polynomial.

Find all real zeros of the function.
f ( x )
 x
3 
8 x
2 
11 x

20

Do Now: Find all real zeros of the function.
f ( x )
 x
3 
4 x
2 
15 x

18

5.6: Find Rational Zeros
HW tonight: p.374 (16-26 even)
Quiz Friday: 5.5, 5.6
(Calculator and no calculator section)

Find all real zeros of the function.
f ( x )

10 x
4 
11 x
3 
42 x
2 
7 x

12

Find all real zeros of the function.
f ( x )

48 x
3 
4 x
2 
20 x

3

Do Now: Find all real zeros of the function.
f ( x )

2 x
4 
5 x
3 
18 x
2 
19 x

42