Section 4.2
Dividing Polynomials
Long Division of Polynomials
Recall: Divided = Quotient*Divisor + Remainder. If the Remainder = 0, then the
quotient is said to be a factor of the dividend.
Example 1: Divide.
12 x 3  x 2  x
a.
3x  1
b.
3x 4  x 3  15
x2  5
Section 4.2 – Dividing Polynomials
1
Dividing Polynomials Using Synthetic Division
We can use synthetic division to divide polynomials if the divisor is of the form x  c.
Example 2: Divide.
a.
6 x 4  x 3  10 x 2  9
1
x
3
 x3  2x  3
b.
x2
Section 4.2 – Dividing Polynomials
2
Popper 15 question 1: What do you divide by if you were to use synthetic
division on f ( x ) 
A. 2
x3  1
x2
B. 1
C.  2
D.  1
Popper 15 question 2: Find the quotient of f ( x ) 
A.  7
B. x 2  2x  3
x3  x  1
x2
C. x 2  2 x  3 D. 7
The Remainder Theorem
If the polynomial P(x) is divided by x – c, then the remainder is P(c).
Example 3: Use synthetic division and the Remainder Theorem to evaluate P(-4) for
P( x)  2 x 3  2 x 2  11x  100
Popper 15 question 3: Find the remainder f ( x ) 
A. 1
B. 0
C.  1
Section 4.2 – Dividing Polynomials
4x  3
.
2x  1
D. 2
3
Example 4: Use synthetic division and the Remainder Theorem to evaluate P(2) for
P ( x)  4 x 4  9 x 3  8 x 2  5 x  2
Example 5: Determine if (x + 2) is a factor of P( x )  x 3  6x 2  3x  10 .
Popper 15 question 4: Given P( x )  x ( x  4) 2 ( x  8)3 , what is the y intercept?
A. to big
B. 4
C 8
Section 4.2 – Dividing Polynomials
D. 0
4
Popper 15 question 5: Given P( x )  x ( x  4) 2 ( x  8)3 , what is the end behavior?
A.
B.
C.
D.
Solutions to Poppers 13 and 14:
Popper 13 question 1: Find the domain: f ( x )  4  2x
A. 2,   B.  ,2 C. 2,   D.  ,2
 2x if

Popper 13 question 2: Calculate: f(-1) if f ( x )  
 x 2 if

A.  2
B. 1
C. 2
x 1
x 1
D. 0
Popper 13 question 3: C varies inversely with the square of w. If w = -2, then C
= 6. Find the constant of proportionality.
3
3
A.  24
B. 24
C
D.
2
2
Popper 13 question 4: State the minimum or maximum of f ( x )  3x 2  6x  5
A. min of -2
B. max of 8 C. max of -2
D. min of 8
Popper 13 question 5: Given f ( x )  x 2  1 and g ( x ) 
A. Not defined
B.
3
4
Section 4.2 – Dividing Polynomials
C. 0
D. 
1
. Find f  g  1 .
2x
3
4
5
Math 1310 Popper 14
Because of Test 3 and this is test 4 material I’m trying to help students. Yes bubble 1-5
all A. This is the best way I know to help with free resposne questions.
Graph the following functions then.
P( x )  x ( x  4) 2 ( x  8 ) 3 Clearly label x-intercepts, y-intercepts. Show the correct
end behavior and the correct behavior at each x –intercept.
Section 4.2 – Dividing Polynomials
6