### Chapter 10 Section 6

```10-6 Dividing Polynomials
Preview
Warm Up
California Standards
Lesson Presentation
10-6 Dividing Polynomials
Warm Up
Divide.
1. m2n ÷ mn4
2. 2x3y2 ÷ 6xy
3. (3a + 6a2) ÷ 3a2b
Factor each expression.
4. 5x2 + 16x + 12
5. 16p2 – 72p + 81
10-6 Dividing Polynomials
California
Standards
10.0 Students add, subtract, multiply, and
divide monomials and polynomials. Students
solve multistep problems, by using these
techniques.
12.0 Students simplify fractions with
polynomials in the numerator and denominator
by factoring both and reducing them to the
lowest terms.
10-6 Dividing Polynomials
To divide a polynomial by a monomial, you can
first write the division as a rational expression.
Then divide each term in the polynomial by the
monomial.
10-6 Dividing Polynomials
Additional Example 1: Dividing a Polynomial by a
Monomial
Divide (5x3 – 20x2 + 30x) ÷ 5x
Write as a rational
expression.
Divide each term in the
polynomial by the
monomial 5x.
Divide out common
factors.
x2 – 4x + 6
Simplify.
10-6 Dividing Polynomials
Check It Out! Example 1a
Divide.
(8p3 – 4p2 + 12p) ÷ (–4p2)
Write as a rational
expression.
Divide each term in the
polynomial by the
monomial –4p2.
Divide out common
factors.
Simplify.
10-6 Dividing Polynomials
Check It Out! Example 1b
Divide.
(6x3 + 2x – 15) ÷ 6x
Write as a rational
expression.
Divide each term in the
polynomial by the
monomial 6x.
Divide out common
factors in each term.
Simplify.
10-6 Dividing Polynomials
Division of a polynomial by a binomial is similar
to division of whole numbers.
10-6 Dividing Polynomials
Additional Example 2A: Dividing a Polynomial by a
Binomial
Divide.
Factor the numerator.
Divide out common factors.
x+5
Simplify.
10-6 Dividing Polynomials
Additional Example 2B: Dividing a Polynomial by a
Binomial
Divide.
Factor both the numerator and
denominator.
Divide out common factors.
Simplify.
10-6 Dividing Polynomials
Put each term of the numerator over the
denominator only when the denominator is a
monomial. If the denominator is a polynomial,
try to factor first.
10-6 Dividing Polynomials
Check It Out! Example 2a
Divide.
Factor the numerator.
Divide out common factors.
k+5
Simplify.
10-6 Dividing Polynomials
Check It Out! Example 2b
Divide.
Factor the numerator.
Divide out common factors.
b–7
Simplify.
10-6 Dividing Polynomials
Check It Out! Example 2c
Divide.
Factor the numerator.
Divide out common factors.
s+6
Simplify.
10-6 Dividing Polynomials
Recall how you used long division to divide
whole numbers as shown at right. You can
also use long division to divide polynomials.
An example is shown below.
(x2 + 3x + 2) ÷ (x + 2)
Divisor
x+1
x + 2) x2 + 3x + 2
x2 + 2x
x+2
x+2
0
Quotient
Dividend
10-6 Dividing Polynomials
Using Long Division to Divide a Polynomial by a Binomial
Step 1 Write the binomial and polynomial in standard form.
Step 2 Divide the first term of the dividend by the first term
of the divisor. This the first term of the quotient.
Step 3 Multiply this first term of the quotient by the binomial
divisor and place the product under the dividend, aligning
like terms.
Step 4 Subtract the product from the dividend.
Step 5 Bring down the next term in the dividend.
Step 6 Repeat Steps 2-5 as necessary until you get 0 or
until the degree of the remainder is less than the degree of
the binomial.
10-6 Dividing Polynomials
Additional Example 3A: Polynomial Long Division
(x2 +10x + 21) ÷ (x + 3)
Write in long division
Step 1 x + 3 ) x2 + 10x + 21
form with expressions
in standard form.
x
Step 2 x + 3 ) x2 + 10x + 21 Divide the first term of the
dividend by the first
term of the divisor to get
the first term of the
quotient.
10-6 Dividing Polynomials
Divide using long division.
(x2 +10x + 21) ÷ (x + 3)
x
Step 3 x + 3 ) x2 + 10x + 21
x2 + 3x
x
Step 4 x + 3 ) x2 + 10x + 21
–(x2 + 3x)
0 + 7x
Multiply the first term of
the quotient by the
binomial divisor. Place
the product under the
dividend, aligning like
terms.
Subtract the product
from the dividend.
10-6 Dividing Polynomials
Divide using long division.
x
Step 5 x + 3 ) x2 + 10x + 21
–(x2 + 3x)
7x + 21
Bring down the next term
in the dividend.
x +7
Step 6 x + 3 ) x2 + 10x + 21 Repeat Steps 2-5 as
necessary.
2
–(x + 3x)
7x + 21
–(7x + 21)
The remainder is 0.
0
10-6 Dividing Polynomials
Check: Multiply the answer and the divisor.
(x + 3)(x + 7)
x2 + 7x + 3x + 21
x2 + 10x + 21 
10-6 Dividing Polynomials
When the remainder is 0, you can check your
simplified answer by multiplying it by the divisor.
You should get the numerator.
10-6 Dividing Polynomials
Additional Example 3B: Polynomial Long Division
Divide using long division.
x – 4) x2 – 2x – 8
Write in long division form.
x+ 2
x – 4) x2 – 2x – 8
–(x2 – 4x)
2x – 8
–(2x – 8)
x2 ÷ x = x
Multiply x  (x – 4). Subtract.
0
Bring down the 8. 2x ÷ x = 2.
Multiply 2(x – 4). Subtract.
The remainder is 0.
10-6 Dividing Polynomials
Check: Multiply the answer and the divisor.
(x + 2)(x – 4)
x2 – 4x + 2x – 8
x2 – 2x + 8
10-6 Dividing Polynomials
Check It Out! Example 3a
Divide using long division.
(2y2 – 5y – 3) ÷ (y – 3)
Step 1
y – 3)
Step 2
2y
y – 3) 2y2 – 5y – 3
2y2
– 5y – 3
Write in long division
form with expressions
in standard form.
Divide the first term of the
dividend by the first
term of the divisor to get
the first term of the
quotient.
10-6 Dividing Polynomials
Check It Out! Example 3a Continued
Divide using long division.
(2y2 – 5y – 3) ÷ (y – 3)
Step 3
2y
y – 3) 2y2 – 5y – 3
2y2 – 6y
2y
Step 4
y – 3) 2y2 – 5y – 3
–(2y2 – 6y)
0+ y
Multiply the first term of
the quotient by the
binomial divisor. Place
the product under the
dividend, aligning like
terms.
Subtract the product
from the dividend.
10-6 Dividing Polynomials
Check It Out! Example 3a Continued
Divide using long division.
2y
Bring down the next term
Step 5 y – 3) 2y2 – 5y – 3
in the dividend.
–(2y2 – 6y)
y –3
Step 6
2y + 1
y – 3) 2y2 – 5y – 3
–(2y2 – 6y)
y–3
–(y – 3)
0
Repeat Steps 2–5 as
necessary.
The remainder is 0.
10-6 Dividing Polynomials
Check It Out! Example 3a Continued
Check: Multiply the answer and the divisor.
(y – 3)(2y + 1)
2y2 + y – 6y – 3
2y2 – 5y – 3

10-6 Dividing Polynomials
Check It Out! Example 3b
Divide using long division.
(a2 – 8a + 12) ÷ (a – 6)
a – 6) a2 – 8a + 12
a– 2
a – 6) a2 – 8a + 12
–(a2 – 6a)
–2a + 12
–(–2a + 12)
0
Write in long division form.
a2 ÷ a = a
Multiply a  (a – 6). Subtract.
Bring down the 12. –2a ÷ a = –2.
Multiply –2(a – 6). Subtract.
The remainder is 0.
10-6 Dividing Polynomials
Check It Out! Example 3b Continued
Check: Multiply the answer and the divisor.
(a – 6)(a – 2)
a2 – 2a – 6a + 12
a2 – 8a + 12 
10-6 Dividing Polynomials
Sometimes the divisor is not a factor of the
dividend, so the remainder is not 0. Then
the remainder can be written as a rational
expression.
10-6 Dividing Polynomials
Additional Example 4: Long Division with a
Remainder
Divide (3x2 + 19x + 26) ÷ (x + 5)
x + 5) 3x2 + 19x + 26
3x + 4
x + 5) 3x2 + 19x + 26
–(3x2 + 15x)
4x + 26
–(4x + 20)
6
Write in long division form.
3x2 ÷ x = 3x.
Multiply 3x(x + 5). Subtract.
Bring down the 26. 4x ÷ x = 4.
Multiply 4(x + 5). Subtract.
The remainder is 6.
Write the remainder as a
rational expression using the
divisor as the denominator.
10-6 Dividing Polynomials
Divide (3x2 + 19x + 26) ÷ (x + 5)
Write the quotient with the
remainder.
10-6 Dividing Polynomials
Check It Out! Example 4a
Divide.
m + 3) 3m2 + 4m – 2
Write in long division form.
3m – 5
m + 3) 3m2 + 4m – 2
–(3m2 + 9m)
3m2 ÷ m = 3m.
Multiply 3m(m + 3). Subtract.
Bring down the –2.
–5m ÷ m = –5 .
–5m – 2
–(–5m – 15) Multiply –5(m + 3). Subtract.
13
The remainder is 13.
10-6 Dividing Polynomials
Check It Out! Example 4a Continued
Divide.
Write the remainder as a
rational expression using the
divisor as the denominator.
10-6 Dividing Polynomials
Divide.
Check It Out! Example 4b
y – 3) y2 + 3y + 2
y+ 6
y – 3) y2 + 3y + 2
–(y2 – 3y)
6y + 2
–(6y –18)
20
y+6+
Write in long division form.
y2 ÷ y = y.
Multiply y(y – 3). Subtract.
Bring down the 2. 6y ÷ y = 6.
Multiply 6(y – 3). Subtract.
The remainder is 20.
Write the quotient with the
remainder.
10-6 Dividing Polynomials
Sometimes you need to write a placeholder
for a term using a zero coefficient. This is
best seen if you write the polynomials in
standard form.
10-6 Dividing Polynomials
Additional Example 5: Dividing Polynomials That
Have a Zero Coefficient
Divide (x3 – 7 – 4x) ÷ (x – 3).
(x3 – 4x – 7) ÷ (x – 3)
x – 3) x3 + 0x2 – 4x – 7
x2
x – 3) x3 + 0x2 – 4x – 7
–(x3 – 3x2)
3x2 – 4x
Write the polynomials in
standard form.
Write in long division form.
Use 0x2 as a placeholder for
the x2 term.
x3 ÷ x = x2
Multiply x2(x – 3). Subtract.
Bring down –4x.
10-6 Dividing Polynomials
x2 + 3x + 5
x – 3) x3 + 0x2 – 4x – 7
–(x3 – 3x2)
3x2 – 4x
–(3x2 – 9x)
5x – 7
–(5x – 15)
8
(x3 – 4x – 7) ÷ (x – 3) =
3x3 ÷ x = 3x
Multiply x2(x – 3). Subtract.
Bring down –4x.
Multiply 3x(x – 3). Subtract.
Bring down – 7.
Multiply 5(x – 3). Subtract.
The remainder is 8.
10-6 Dividing Polynomials
Remember!
Recall from Chapter 7 that a polynomial in one
variable is written in standard form when the
degrees of the terms go from greatest to least.
10-6 Dividing Polynomials
Check It Out! Example 5a
Divide (1 – 4x2 + x3) ÷ (x – 2).
(x3 – 4x2 + 1) ÷ (x – 2)
x – 2) x3 – 4x2 + 0x + 1
x2 – 2x – 4
x – 2) x3 – 4x2 + 0x + 1
–(x3 – 2x2)
– 2x2 + 0x
–(–2x2 + 4x)
– 4x + 1
–(–4x + 8)
–7
Write in standard form.
Write in long division form.
Use 0x as a placeholder for
the x term.
x3 ÷ x = x2
Multiply x2(x – 2). Subtract.
Bring down 0x. – 2x2 ÷ x = –2x.
Multiply –2x(x – 2). Subtract.
Bring down 1.
Multiply –4(x – 2). Subtract.
10-6 Dividing Polynomials
Check It Out! Example 5a Continued
Divide (1 – 4x2 + x3) ÷ (x – 2).
(1 – 4x2 + x3) ÷ (x – 2) =
10-6 Dividing Polynomials
Check It Out! Example 5b
Divide (4p – 1 + 2p3) ÷ (p + 1).
(2p3 + 4p – 1) ÷ (p + 1)
p + 1) 2p3 + 0p2 + 4p – 1
2p2 – 2p + 6
p + 1) 2p3 + 0p2 + 4p – 1
–(2p3 + 2p2)
– 2p2 + 4p
–(–2p2 – 2p)
6p – 1
–(6p + 6)
–7
Write in standard form.
Write in long division form.
Use 0p2 as a placeholder
for the p2 term.
p3 ÷ p = p2
Multiply 2p2(p + 1). Subtract.
Bring down 4p. –2p2 ÷ p = –2p.
Multiply –2p(p + 1). Subtract.
Bring down –1.
Multiply 6(p + 1). Subtract.
10-6 Dividing Polynomials
Check It Out! Example 5b Continued
(2p3 + 4p – 1) ÷ (p + 1) =
10-6 Dividing Polynomials
Lesson Quiz: Part I
1. (12x2 – 4x2 + 20x) ÷ 4x
2.
2x + 3
3.
x–2
4.
x+3
3x2 – x + 5
10-6 Dividing Polynomials
Lesson Quiz: Part II
Divide using long division.
5. (x2 + 4x + 7)  (x + 1)
6. (8x2 + 2x3 + 7)  (x + 3)
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