Download Connections Dividing Polynomials

Dividing Polynomials
Connections
Have you ever wondered . . .
• How cryptographers create and break codes?
• How engineers determine acceleration of an engine?
• How temperature control systems are designed?
All of these areas involve dividing polynomials. Dividing polynomials is a stepping stone to higher mathematics.
Dividing polynomials is similar to dividing large numbers. When you divide large numbers,
you must make sure that your place values align. When you divide polynomials, like terms
must align.
Line up like terms.
A term can have two parts: the coefficient and the variable. Divide both the coefficients and
the variables. When you divide variables, subtract the exponents.
Six divided by two is three; x2 divided by x is x.  6x2 ÷ 2x = 3x
The simplest division of polynomials is finding and factoring out a monomial (one-term)
greatest common factor (GCF). That’s the largest term that can be divided out of each term
of a polynomial.
2x2 is the greatest common factor.  6x3 + 4x2
Factor out the greatest common factor.  (2x2)(3x + 2)
185
Essential Math Skills
Learn
It!
Long Division with Polynomials
To find a greatest common factor (GCF), look at all terms in the polynomial.
• What number can be divided into all the coefficients?
• What variables do they all have in common?
• What is the lowest power of the common variables?
The GCF will be the largest factor of the coefficients and the common variables to the lowest
power. Divide each term by the GCF to factor the polynomial.
Polynomial
GCF
Factored Form
3x2 + 6xy + 9x2y
3x
3x(x + 2y + 3xy)
24x5 - 8x3 + 4x2
4x2
4x2(6x3 - 2x + 1)
7y2 + 14y3 - 17y
y
y(7y + 14y2 - 17)
You can use the GCF to divide a polynomial by a binomial. The format and steps are just like
long division. Instead of keeping track of place values, you will line up like terms.
A marketing executive comes up with this model for the returns on advertising
dollars in a specific market, where x equals the amount spent on advertising.
7x3 + 3x2 - 10
x-1
Simplify this expression by dividing 7x3 + 3x2 - 10 by x - 1.
Write the Long Division Problem
First, write the problem as long division. Be sure your terms are ordered from highest exponent to lowest exponent. Because each term is a different “place value,” be sure that you
are not missing terms in your dividend or divisor. If you are, use zero as a placeholder. For
example, in this problem, the dividend has x3 and x2 but no x. Use 0x as a placeholder.
?
1. Write the long division problem.
There is a missing term in the dividend, the first power or x term. Keep the dividend and
divisor in descending order, and insert 0x for the missing term.
x - 1g 7x3 + 3x2 + 0x - 10
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Dividing Polynomials
Divide the First Terms
Because the terms are in descending order, you can start by just looking at the first terms.
• Divide the first term of the divisor into the first term of the dividend.
• Write the result above the equation, lining up common terms.
• Multiply the whole divisor by that expression.
• Write the result below the dividend, lining up common terms.
• Subtract, then bring down the next term.
?
2.Divide x - 1 into the terms with the largest powers.
Begin by dividing the first terms in each polynomial. 7x3 divided by x is 7x2.
7x 2
2
Multiply 7x2 times x - 1. Write the result 7x3 - 7x2  x - 1g 7x + 3x + 0x - 10
- (7x3 - 7x2)
below the dividend, lining up common terms. 
Subtract 7x3 - 7x2 from 7x3 + 3x2 to get 10x2. 
10x2 + 0x
Write 7x2 above 3x2 to align terms. 
3

Bring down 0x.
Repeat
Repeat the division until you have reached the end of the polynomial.
?
3. Complete the division.
Start by dividing x into 10x2 and continue to the end of
the polynomial.
The quotient is 7x2 + 10x + 10.
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Essential Math Skills
e
ic
Pract
It!
Use your knowledge of factoring and dividing polynomials to answer the
following questions.
1. -15x3y + 9x2y2 - 6x2
Find the greatest common factor.
2. 8x3 + 48x2 + 12x - 34
Find the greatest common factor.
3. 45x2 - 18x + 9
Find the greatest common factor.
4. A business revenue model is represented by the expression
15x3 - 3x2 + 924x. If the revenue is a multiple of sales, and sales are
3x, what is the revenue per sale?
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Dividing Polynomials
5. The area of a garden measures 3x + 5 feet by 3x + 3 feet. How many plots with an area
of x + 1 square feet will fit in the garden?
6. The area of a rectangle is 4x3 - 17x2 + 8x - 1, and its length is
4x - 1. Find an expression for its width.
Area = length ◊ width
Area = 4x3 - 17x2 + 8x - 1
length = 4x - 1
7. (15y2 - 15) ÷ (3y + 3)
a. Divide the polynomials.
b. What error might you make in dividing the polynomials? How could you avoid it?
8. A financial model is represented by the following expression.
x3 + 2x2 - 4x - 8
x-2
Simplify this expression.
Math Tip
Use placeholder
terms with
coefficients of zero
when terms are
missing.
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Essential Math Skills
9. 2x2 - 3g 4x 4 + 6x2 - 18
a. Divide the polynomials.
b. What error might you make in dividing the polynomials? How could you avoid it?
10. Jim turned in the following division solution.
What mistake did Jim make? What advice would you give Jim to
avoid this mistake in the future?
Using
Un P A C
To check your
factors, multiply
them. The result
should be the
original polynomial.
derstand
lan
ttack
heck
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Dividing Polynomials
Check Your Skills
Use your knowledge of factoring and dividing polynomials to answer the following questions.
5 2
2
2 2
2 3
1. 3x y - 15x y + 42x y - 14x y
What is the greatest common factor of this polynomial?
a. x2y
b.
3x2y
c.
xy
d. 3xy2
5
3
2
2. 24x + 18x - 30x
Which expression shows the factors of this polynomial?
a. 6x2(4x3 + 3x - 5)
b. 8x(3x4 + 2x2 - 4x)
c. 3x2(8x3 + 6x - 10)
d. 3x(8x4 + 6x2 - 10x)
3
2
2
3. A container has a volume modeled by this expression: 30xy - 40y + 15xy - 25x y
How many objects with a volume of 5y would fit in the container?
4
+ 2+ 4. 2x 1g 30x 3x x 4
What is the first term in the quotient of this division problem?
a. 15x
b. 15x3
c.
30x
d. 30x4
5. A box has a height of x, a length of x + 2 inches and a width of 3x - 5 inches. How many
objects with a volume of x can fit in the box?
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Essential Math Skills
6. (5y2 + 6y - 27) ÷ (y + 3)
What is the quotient of this division problem?
a. 5y + 9
b. 3y + 2
c. 3y - 2
d. 5y - 9
7. The foundation of a building has an area of 2x2 + 11x + 12. The architect wants to divide
the foundation into regions with an area of 2x + 3 in order to place reinforcements. How
many regions will the foundation have?
a. x2- 4
b. x - 4
c. x2+ 4
d. x + 4
8.A rectangle has an area of x3 - 6x2 + x - 6. Its width is x - 6. What is its length?
a.
x2 + 1
b.
x - 1
c.
x2 + 1
d.
x + 1
9. (2x2 + 12x - 14) ÷ (2x - 2)
Divide.
Remember
the Concept
10.
Gas mileage as a factor of speed for a particular car is
-0.0004s2 + 0.12s - 5. A new type of hybrid vehicle has a
gas mileage of -0.02s + 1. What is the ratio of the two
vehicles’ gas mileage as a factor of speed?
To find the GCF, look for
the biggest numerical
factor and the biggest
variable factor.
Divide just as with long
division, treating each
term as a different
“place value.”
For each division step,
divide the first terms.
192
Answers and Explanations
Dividing Polynomials
page 185
7a.3y + 3
Long Division with Polynomials
Practice It!
pages 188–190
g
5y - 5
2
15y + 0y - 15
- (15y2 + 15y)
- 15y - 15
1.
GCF = 3x2
- (- 15y - 15)
-15x3y + 9x2y2 - 6x2 = 3x2(-5xy + 3y2 - 2)
2.
GCF = 2
8x + 48x + 12x - 34 = 2(4x + 24x + 6x - 17)
3
2
3
2
3.
GCF = 9
45x2 - 18x + 9 = 9(5x2 - 2x + 1)
4.
3x + 0
g
2
- x + 308
5x
2
15x - 3x + 924x + 0
3
0
7b.You might forget to add 0y to the polynomial to
divide. You can avoid this error by checking the terms
in your polynomial before dividing.
8.
x - 2g
x 2 + 4x + 4
x + 2x 2 - 4x - 8
3
3
2
- ( x - 2x )
4x 2 - 4x
- (15x 3 + 0x 2)
2
- (4x - 8x)
4x - 8
- 3x 2 + 924x
- ( - 3x 2 +
- (4x - 8)
0
0x)
924x + 0
- (924x + 0)
0
9a.
5.
Multiply to find the area:
(3x + 5)(3x + 3) = 9x2 + 24x + 15
Divide to find the plots that will fit:
x+1
g
9x + 15
2
9x + 24x + 15
9b.An error you might make is subtracting 6x2 from 6x2
instead of subtracting -6x2. If you make this error, you
might get stuck. One way to avoid it is to distribute
the subtraction sign to the polynomial that you’re
subtracting.
- (9x 2 + 9x)
15x + 15
- (15x + 15)
0
6.
4x - 1
g
2
x - 4x + 1
2
4x - 17x + 8x - 1
3
- (4x 3 -
10.Jim did not use placeholders for missing terms. His
terms aren’t aligned, and he has the wrong solution.
You might advise him to check the problem vertically
to make sure the terms align.
2
x)
- 16x 2 + 8x
- (- 16x 2 + 4x)
4x - 1
- (4x - 1)
0
i
Essential Math Skills
Check Your Skills
pages 191–192
1.
a. x2y
8.
c. x2 + 1
x - 6
2.
a. 6x2(4x3 + 3x - 5)
3.
6xy2 - 8y + 3x - 5x2
g
2
1
x +
2
x - 6x + x - 6
3
- (x 3 - 6 x 2 )
0+x-6
4.
b. 15x3
- (x - 6)
5.
3x2 + x - 10
0
(x)(x + 2)(3x - 5) ÷ x
9.
x+7
(x)(3x2 + x - 10) ÷ x
6.
d. 5y - 9
y + 3
g
2x - 2
5y - 9
2
5y + 6y - 27
g
x+ 7
2
2x + 12x - 14
- (2x 2 - 2x)
14x - 14
- (5y2 + 15y)
- (14x - 14)
- 9y - 27
- (- 9y - 27)
0
10.0.02s - 5
0
7.
d. x + 4
2x + 3
ii
g
x+ 4
2
2x + 11x + 12
- (2x 2 + 3x)
- 0.0004s 2 + 0.12s - 5
- 0.02s + 1
- 0.02s + 1
g
0.02s - 5
- 0.0004s 2 + 0.12s - 5
- (- 0.0004s 2 + 0.02s)
8x + 12
0. 1 s - 5
- (8x + 12)
- (0.1s - 5)
0
0