8-5
Dividing Polynomials
TEKS FOCUS
VOCABULARY
ĚSynthetic division – Synthetic division is a process for dividing a
TEKS (7)(C) Determine the quotient of a
polynomial of degree three and degree four
when divided by a polynomial of degree one
and of degree two.
polynomial by a linear expression x - a. You list the standard-form
coefficients (including zeros) of the polynomial, omitting all variables
and exponents. You use a for the “divisor” and add instead of subtract
throughout the process.
TEKS (1)(A) Apply mathematics to problems
arising in everyday life, society, and the
workplace.
ĚApply – use knowledge or information for a specific purpose, such as
solving a problem
Additional TEKS (1)(G)
ESSENTIAL UNDERSTANDING
You can divide polynomials using steps that are similar to the long-division steps
that you use to divide whole numbers.
Key Concept
The Division Algorithm for Polynomials
You can divide polynomial P (x) by polynomial D (x) to get polynomial quotient Q (x)
and polynomial remainder R (x). The result is P (x) = D (x)Q (x) + R (x).
Q (x)
D (x)) P (x)
##
#
R (x)
If R (x) = 0, then P (x) = D (x)Q (x) and D (x) and Q (x) are factors of P (x).
To use long division, P (x) and D (x) should be in standard form with zero coefficients
where appropriate. The process stops when the degree of the remainder, R (x), is less
than the degree of the divisor, D (x).
Theorem
The Remainder Theorem
If you divide a polynomial P (x) of degree n Ú 1 by x - a, then the remainder is P (a).
PearsonTEXAS.com
363
Problem 1
P
Using Polynomial Long Division
Use polynomial long division to divide 4x3 + 3x2 − 20x − 46 by x2 − 5. What is
the quotient and remainder?
x2
-
4x
+
- 20x - 46
4x 3
- 20x
3x2
- 46
5) 4x 3
3
Divide: 4x2 = 4x.
x
Multiply: 4x(x 2 - 5) = 4x 3 - 20x.
Subtract to get 3x 2. Bring down -46.
3x2
Repeat the process of dividing, multiplying, and subtracting.
How can you check
your result?
Show that
(divisor)(quotient) +
remainder = dividend.
4x + 3
x2 - 5) 4x3 + 3x2 - 20x - 46
4x3
- 20x
3x2
- 46
- 15
3x2
-31
2
Divide: 3x2 = 3
x
Multiply: 3(x 2 - 5) = 3x 2 - 15.
Subtract to get - 31.
T quotient is 4x + 3 with remainder -31.
The
You can say: 4x ⫹ 3, R ⫺31.
C
Check
( 2 - 5)(4x + 3) - 31 = (4x3 + 3x2 - 20x - 15) - 31
(x
= 4x3 + 3x2 - 20x - 46 ✔
Multiply (x 2 - 5)(4x + 3).
Simplify.
Problem
bl
2
TEKS Process Standard (1)(G)
Using Polynomial Long Division to Check Factors
A Use polynomial long division to divide P(x) = 3x4 − 4x 3 + 12x 2 + 5 by
x 2 + 1. Is x 2 + 1 a factor of P(x)?
3x 2 - 4x + 9
x2
+ 0x +
1) 3x 4 -
12x 2 + 0x + 5
3x 2
3
-4x + 9x 2 + 0x
-4x 3 + 0x 2 - 4x
9x 2 + 4x + 5
9x 2 + 0x + 9
4x - 4
3x 4 +
4x 3 +
Include 0x terms.
0x 3 +
The degree of the remainder is
less than the degree of the
divisor. Stop!
The remainder is not zero. x2 + 1 does not divide 3x4 - 4x3 + 12x2 + 5 evenly
and is not a factor of P(x).
continued on next page ▶
364
Lesson 8-5
Dividing Polynomials
Problem 2
Can you use the
Factor Theorem to
help answer this
question?
Yes; recall that if
P(a) = 0, then x - a is
a factor of P(x).
continued
B Is x − 2 a factor of P (x) = x4 − 16? If it is, write P (x) as a product of two factors.
Step 1 Use the Factor Theorem to determine if x - 2 is a factor of x4 - 16.
S
P (2) = 24 - 16
= 16 - 16
=0
Since P (2) = 0, x - 2 is a factor of P (x).
Step 2 Use polynomial long division to find the other factor.
x3 + 2x2 + 4x + 8
x-
2) x4
x4
+ 0x3 + 0x2 + 0x - 16
- 2x3
2x3 + 0x2
2x3 - 4x2
4x2 + 0x
4x2 - 8x
8x - 16
8x - 16
0
P (x) = (x - 2)(x 3 + 2x 2 + 4x + 8)
Problem
P
bl
3
Using Synthetic Division
To divide by x + 2
what number do you
use for the synthetic
divisor?
x + 2 = x - (-2), so
use - 2.
U synthetic division to divide x4 − 14x3 + 51x2 − 54x − 110 by x + 2.
Use
What
is the quotient and remainder?
W
Step 1 Reverse the sign of +2. Write
S
the coefficients of the polynomial.
-2 ; 1 -1 4
5 1 -5 4 -1 1 0
Step 2 Bring down the first coefficient.
-2 ; 1
-14
51
-54 -110
1
Step 3 Multiply the coefficient by the divisor.
Add to the next coefficient.
-2 ; 1
1
-14
-2
-16
51
-54 -110
Step 4 Continue multiplying and adding
through the last coefficient.
-2 ; 1 -14 51 -54 -110
-2 32 -166 440
1 -16 83 -220 330
The quotient is x3 - 16x2 + 83x - 220, R 330.
PearsonTEXAS.com
365
Problem 4
P
TEKS Process Standard (1)(A)
Using Synthetic Division to Solve a Problem
oblem
Crafts The polynomial x3 + 7x2 − 38x − 240
expresses the volume, in cubic inches, of the
shadow box shown.
A What are the dimensions of the box?
How can you use the
picture to help solve
the problem?
The picture gives the width
of the box. Remember
for a rectangular prism,
V = / * w * h.
(Hint: The length is greater than the
height (or depth).)
-5 ; 1
7 -38 -240
240
-5 -10
1
2 -48
0
x2 + 2x - 48 = (x - 6)(x + 8)
So, x3 + 7x2 - 38x - 240 = (x + 5)(x2 + 2x - 48)
= (x + 5)(x - 6)(x + 8)
x+5
The length, width, and height (or depth) of the box are
(x + 8) in., (x + 5) in., and (x - 6) in., respectively.
B If the width of the box is 15 in., what are the other
two dimensions?
The width of the box is x + 5. So if x + 5 = 15, then x = 10.
Substitute for x to find the length and height (or depth).
Length: x + 8 = 10 + 8 = 18 in.
Height: x - 6 = 10 - 6 = 4 in.
Problem
bl
5
Evaluating a Polynomial
Is there a way to
find P(3) without
substituting?
Use synthetic division.
P(3) is the remainder.
Given that P (x) = x4 − 2x2 − x + 122, what is P (3)?
G
By
B the Remainder Theorem, P (3) is the remainder when you
divide P (x) by x - 3.
d
3; 1 0
3
1 3
P (3) = 182.
366
Lesson 8-5
Dividing Polynomials
-2
-9
7
-1 122
21 60
20 182
1 8 2
. . . . . . .
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0 0
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6
7 7 7
8 8 8
9 9 9
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Divide using long division. Check your answers.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. 1 3x2 + 7x - 20 2 , (x + 4)
2. 1 x3 + 3x2 - x + 2 2 , (x - 1)
3. 1 2x3 - 3x2 - 18x - 8 2 , (x - 4)
4. 1 x3 + 5x2 - 4x - 20 2 , (x2 - 4)
Divide.
5. 1 2x3 + 9x2 + 14x + 5 2 , (2x2 + 1)
6. 1 x4 + 3x2 + x + 4 2 , (x + 3)
7. 1 x4 + 4x3 - x - 4 2 , (x2 - 1)
8. 1 3x4 - 5x3 + 2x2 + 3x - 2 2 , (3x - 2)
Determine whether each binomial is a factor of x3 + 4x2 + x − 6.
9. x + 1
10. x + 2
11. x + 3
12. x - 3
Divide using synthetic division.
13. 1 x3 + 3x2 - x - 3 2 , (x - 1)
14. 1 x3 - 4x2 + 6x - 4 2 , (x - 2)
15. 1 x3 - 7x2 - 7x + 20 2 , (x + 4)
16. 1 x3 - 3x2 - 5x - 25 2 , (x - 5)
17. 1 x2 + 3 2 , (x - 1)
18. 1 3x3 + 17x2 + 21x - 9 2 , (x + 3)
19. 1 x3 + 27 2 , (x + 3)
20. 1 6x2 - 8x - 2 2 , (x - 1)
Use synthetic division and the given factor to completely factor
each polynomial function.
21. y = x3 + 2x2 - 5x - 6; (x + 1)
22. y = x3 - 4x2 - 9x + 36; (x + 3)
23. Apply Mathematics (1)(A) The volume, in cubic inches, of the decorative
box shown can be expressed as the product of the lengths of its sides as
V (x) = x3 + x2 - 6x. What linear expressions with integer coefficients represent
the length and height of the box?
x
Use synthetic division and the Remainder Theorem to find P (a).
24. P (x) = x3 + 4x2 - 8x - 6; a = -2
25. P (x) = x3 + 4x2 + 4x; a = -2
26. P (x) = x3 - 7x2 + 15x - 9; a = 3
27. P (x) = x3 + 7x2 + 4x; a = -2
28. P (x) = 6x3 - x2 + 4x + 3; a = 3
29. P (x) = 2x3 - x2 + 10x + 5; a = 12
30. P (x) = 2x3 + 4x2 - 10x - 9; a = 3
31. P (x) = 2x4 + 6x3 + 5x2 - 45; a = -3
PearsonTEXAS.com
367
32. Select Techniques to Solve Problems (1)(C) Your friend multiplies x + 4 by a
quadratic polynomial and gets the result x3 - 3x2 - 24x + 30. The teacher says
that everything is correct except for the constant term. Find the quadratic
polynomial that your friend used. What is the correct result of multiplication?
33. Display Mathematical Ideas (1)(G) A student used synthetic division
to divide x3 - x2 - 2x by x + 1. Describe and correct the error shown.
1
34. Connect Mathematical Ideas (1)(F) When a polynomial is divided
by (x - 5), the quotient is 5x2 + 3x + 12 with remainder 7. Find the
polynomial.
35. Apply Mathematics (1)(A) The expression 13(x3 + 5x2 + 8x + 4) represents
the volume of a square pyramid. The expression x + 1 represents the height
of the pyramid. What expression represents the side length of the base?
(Hint: The formula for the volume of a pyramid is V = 13Bh.)
36. Analyze Mathematical Relationships (1)(F) Divide. Look for patterns in
your answers.
a. 1 x2 - 1 2 , (x - 1)
b. 1 x3 - 1 2 , (x - 1)
c. 1 x4 - 1 2 , (x - 1)
d. Using the patterns, factor x5 - 1.
37. Select Tools to Solve Problems (1)(C) The remainder from the division of the
polynomial x3 + ax2 + 2ax + 5 by x + 1 is 3. Find a.
38. Use synthetic division to find (x2 + 4) , (x - 2i).
39. Display Mathematical Ideas (1)(G) Suppose 3, -1, and 5 are zeros of a cubic
polynomial function f (x). What is the sign of f (1) f (4)? (Hint: Sketch the
graph; consider all possibilities.)
#
TEXAS Test Practice
T
40. What is the remainder when x2 - 5x + 7 is divided by x + 1?
A. 1
B. 3
C. 11
D. 13
41. What is the least degree of a polynomial that has a zero of multiplicity 3 at 1,
a zero of multiplicity 1 at 0, and a zero of multiplicity 2 at 2?
F. 3
G. 4
H. 5
J. 6
42. The equation y = 0.17x relates your weight on the Moon y to your weight on Earth
x in pounds. If Al weighs 130 lb on Earth, what would he weigh on the Moon?
A. 22.1 lb
B. 92.3 lb
C. 130 lb
pr 2.
D. 764.7 lb
43. The formula for the area of a circle is A =
Solve the equation for r. If the
area of a circle is 78.5 cm2, what is the radius? Use 3.14 for p.
368
Lesson 8-5
Dividing Polynomials
1
1
-1
1
0
-2
0
-2