1 topic 8.2.2 dividing rational expressions dividing rational expressions
TRANSCRIPT
2
Lesson
1.1.1
California Standard:13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
What it means for you:You’ll divide rational expressions by factoring and cancelling.
Dividing Rational ExpressionsDividing Rational ExpressionsTopic
8.2.2
Key words:• rational• reciprocal• common factor
3
Lesson
1.1.1
Dividing by rational expressions is a lot like multiplying — you just have to do an extra step first.
Topic
8.2.2
That extra step is finding the reciprocal.
Dividing Rational ExpressionsDividing Rational Expressions
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Lesson
1.1.1
Dividing is the Same as Multiplying by the Reciprocal
Topic
8.2.2
Given any nonzero expressions m, c, b, and v:
m
c÷ = • =
b
v
v
b
m
c
mv
cb
That is, to divide by , multiply by the reciprocal of .m
c
b
v
b
v
Dividing Rational ExpressionsDividing Rational Expressions
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Lesson
1.1.1
Dividing is the Same as Multiplying by the Reciprocal
Topic
8.2.2
You can extend this concept to the division of any rational expression.
The question you’re trying to answer is…
“How many times does go into 10?”
…or “How many halves are in 10?”
1
2
Suppose you pick a number such as 10 and divide by . 1
2
Dividing Rational ExpressionsDividing Rational Expressions
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10 ÷ = 20 10 × 2 = 20
10 ÷ = 30 10 × 3 = 30
10 ÷ = 40 10 × 4 = 40
10 ÷ = 10n 10 × n = 10n
12
12
12
12
Lesson
1.1.1
Dividing is the Same as Multiplying by the Reciprocal
Topic
8.2.2
Division Equivalent to
So, 10 divided by a fraction is equivalent to 10 multiplied by the reciprocal of that fraction.
Dividing Rational ExpressionsDividing Rational Expressions
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Lesson
1.1.1
Dividing is the Same as Multiplying by the Reciprocal
Topic
8.2.2
Dividing anything by a rational expression is the same as multiplying by the reciprocal of that expression.
So you can always rewrite an expression a ÷ b in the
form a • = (where b is any nonzero expression).1
b
a
b
As always, you should cancel any common factors in your answer to give a simplified fraction.
Dividing Rational ExpressionsDividing Rational Expressions
8
Example 1
Solution follows…
Topic
8.2.2
Solution
Rewrite the division as multiplication by the reciprocal of the divisor.
Simplify ÷ (k + 5).k2 – 25
2k
÷ (k + 5) can be written as:k2 – 25
2k÷
k2 – 25
2k
k + 5
1
=k2 – 25
2k
1
k + 5•
Factor as much as you can: =(k – 5)(k + 5)
2k
1
k + 5•
Solution continues…
Dividing Rational ExpressionsDividing Rational Expressions
9
Example 1
Topic
8.2.2
Solution (continued)
Check your answer. Multiply your answer by (k + 5):
Simplify ÷ (k + 5).k2 – 25
2k
Cancel any common factors between the numerators and denominators.
=(k – 5)(k + 5)
2k
1
k + 5•
1
1
=k – 5
2k
=k – 5
2k
(k – 5)(k + 5)
2k
k2 – 25
2k•
k + 5
1=
Dividing Rational ExpressionsDividing Rational Expressions
10
Cancel any common factors between the numerators and denominators.
Example 2
Topic
8.2.2
Solution
Rewrite the division as multiplication by the reciprocal of the divisor.
Solution follows…
Factor all numerators and denominators.
Simplify ÷ .m2 – 4
m2 – 3m + 2
2m
m – 1
m2 – 4
m2 – 3m + 2
m – 1
2m•=
(m + 2)(m – 2)
(m – 2)(m – 1)
m – 1
2m•=
m + 2
2m=
1
1
1
1
Dividing Rational ExpressionsDividing Rational Expressions
11
Divide and simplify each expression.
1. 2.
3. 4.
5.
Lesson
1.1.1
Guided Practice
Topic
8.2.2
Solution follows…
1
b2c2d2
bdc3
bcd2
abc÷
a2 – 9
a2 + a – 6
a + 3
a – 2÷
a(b – 2)
b + 1
2a
(b + 1)(b – 1)÷
a2 + 3a + 2
a2 – a – 6
a2 – 1
a2 – 4a + 3÷
x2 – 5x – 6
x2 + 3x – 10
x2 – 4x – 5
x2 – 25÷
abc
d3
b2 – 3b + 2
2
a – 3
a + 3
x – 6
x – 2
Dividing Rational ExpressionsDividing Rational Expressions
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Lesson
1.1.1
You Can Divide Long Strings of Expressions At Once
Topic
8.2.2
Just like multiplication, you can divide any number of rational expressions at once, but it makes a big difference which order you do things in.
If there are no parentheses, you always work through the calculation from left to right, so that:
a
b÷ ÷
c
d
e
f
a
b= • ÷
d
c
e
f
a
b= • •
d
c
f
e
Dividing Rational ExpressionsDividing Rational Expressions
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Rewrite each division as a multiplication by the reciprocal of the divisor.
Example 3
Topic
8.2.2
Solution
Solution follows…
Simplifyx2 + 5x + 6
x2 + 3x
x2 + x – 2
2x2 + 2x÷
x2 + 2x + 1
x – 1÷ .
=x2 + 5x + 6
x2 + 3x
2x2 + 2x
x2 + x – 2•
x2 + 2x + 1
x – 1÷
=x2 + 5x + 6
x2 + 3x
2x2 + 2x
x2 + x – 2•
x – 1
x2 + 2x + 1•
Solution continues…
Dividing Rational ExpressionsDividing Rational Expressions
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Example 3
Topic
8.2.2
Solution (continued)
Factor all numerators and denominators.
Cancel any common factors between the numerators and denominators.
Simplifyx2 + 5x + 6
x2 + 3x
x2 + x – 2
2x2 + 2x÷
x2 + 2x + 1
x – 1÷ .
=x2 + 5x + 6
x2 + 3x
2x2 + 2x
x2 + x – 2•
x – 1
x2 + 2x + 1•
=(x + 2)(x + 3)
x(x + 3)• •
2x(x + 1)
(x + 2)(x – 1)
x – 1
(x + 1)(x + 1)=
2
x + 11
1 1
11
1 1
1
1
1
Dividing Rational ExpressionsDividing Rational Expressions
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Lesson
1.1.1
You Can Divide Long Strings of Expressions At Once
Topic
8.2.2
Parentheses override this order of operations, so you need to simplify any expressions in parentheses first:
a
b= ÷
c • f
d • e
a
b= •
d • e
c • f
a
b÷ ÷
c
d
e
f
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Divide and simplify each expression.
6.
7.
8.
Lesson
1.1.1
Guided Practice
Topic
8.2.2
Solution follows…
k2 – 1
2k2 – 14k
k2 + 5k – 6
k2 – 9k + 14÷
–k2 + 3k – 2
2k2 – 10k÷
x2 – 4x – 12
2x2 – 3x – 2
–x2 + 2x + 8
3x3 + 3x2 – 18x÷
6x3 – 36x2
–2x2 + 7x + 4÷
(x – 2)
(x + 3)
(x – 2)(x + 4)
(x + 3)÷ ÷
(x + 4)
(x + 1)
x + 3
2x
1
x + 1
k2 – 4k – 5
k2 + 5k – 6–
Dividing Rational ExpressionsDividing Rational Expressions
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Lesson
1.1.1
You Can Multiply and Divide at the Same Time
Topic
8.2.2
Say you have an expression like this to simplify:
Again, you work from left to right, and anywhere you get a division, multiply by the reciprocal, so:
a
b÷ ×
c
d
e
f
a
b÷ ×
c
d
e
f
a
b× ×
d
c
e
f=
Dividing Rational ExpressionsDividing Rational Expressions
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Rewrite any divisions as multiplications by reciprocals.
Example 4
Topic
8.2.2
Solution
Solution follows…
Simplifyp2 + pq – 2q2
p2 – 2pq – 3q2
p2 + q2
pq + 2q2×
p2 – 2pq + q2
p2 – 3pq÷ .
=p2 + pq – 2q2
p2 – 2pq – 3q2
pq + 2q2
p2 + q2×
p2 – 2pq + q2
p2 – 3pq×
Factor all numerators and denominators.
=p
q=
(p + 2q)(p – q)
(p – 3q)(p + q)× ×
(p – q)(p + q)
q(p + 2q)
p(p – 3q)
(p – q)(p – q)
Cancel any common factors.
1
1
1
1
1
11
11
1
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Example 5
Topic
8.2.2
Solution
Solution follows…
Justify your work.
Show that2a2 – 7a + 3
a2 + 4a – 21÷
2a – 2
a + 1=
2a2t + at – t
2a2t + 12at – 14t
The question asks you to justify your work, so make sure you can justify all your steps.
Start with left-hand side
Definition of division
2a2 – 7a + 3
a2 + 4a – 21÷
2a2t + at – t
2a2t + 12at – 14t
=2a2 – 7a + 3
a2 + 4a – 21×
2a2t + 12at – 14t
2a2t + at – t
Solution continues…
Dividing Rational ExpressionsDividing Rational Expressions
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=2a2 – 7a + 3
a2 + 4a – 21×
2a2t + 12at – 14t
2a2t + at – t
Example 5
Topic
8.2.2
Solution (continued)
Justify your work.
Show that2a2 – 7a + 3
a2 + 4a – 21÷
2a – 2
a + 1=
2a2t + at – t
2a2t + 12at – 14t
=(2a – 1)(a – 3)
(a + 7)(a – 3)×
2t(a2 + 6a – 7)
t(2a2 + a – 1)
=(2a – 1)(a – 3)
(a + 7)(a – 3)×
2t(a + 7)(a – 1)
t(2a – 1)(a + 1)
Distributive property
Distributive property
Equation carried forward
Solution continues…
Dividing Rational ExpressionsDividing Rational Expressions
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=(2a – 1)(a – 3)
(a + 7)(a – 3)×
2t(a + 7)(a – 1)
t(2a – 1)(a + 1)
=(2a – 1)
(a + 1)×
t(2a – 1)(a + 7)(a – 3)
t(2a – 1)(a + 1)(a – 3)
Example 5
Topic
8.2.2
Solution (continued)
Justify your work.
Show that2a2 – 7a + 3
a2 + 4a – 21÷
2a – 2
a + 1=
2a2t + at – t
2a2t + 12at – 14t
=(2a – 1)
(a + 1)=
2a – 2
a + 1Inverse and identity properties, and distributive property
Commutative and associative properties of multiplication
Equation carried forward
Dividing Rational ExpressionsDividing Rational Expressions
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Simplify these rational expressions.
9.
10.
11.
12.
Lesson
1.1.1
Guided Practice
Topic
8.2.2
Solution follows…
t2 – 1
t2 + 2t – 3
t + 1
t2 + 4t + 3÷
t – 1
1×
a2 + a – 12
a2 + a – 2
a2 + 5a + 4
a2 + 2a + 1÷
a2 + 2a – 3
a2 – 2a – 3×
x2 + 5x – 14
x2 – 4x – 21
x2 + 6x – 7
x2 – 6x – 7÷
x2 + 2x – 3
x2 – 5x + 6×
a2 – 1
a2 – 4
a2 – 2a – 3
a2 – 3a – 10÷
a2 – 5a + 6
–a2 + 2a + 15×
a + 3
a + 2
x + 1
x – 3
t2 – 1
a – 1
a + 3–
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Independent Practice
Solution follows…
Topic
8.2.2
Divide and simplify each expression.
1. 2.
3. 4.
5. 6.
b2 + b – 2
k2 – m2
2k2 + km – m2
2k2 + 2m
2k2 + 3km – 2m2÷
t2 + 2t – 3
t2 + 4t + 3
3t – 3
t2 – t – 2÷
–x3 – 3x2 – 2x
x2 – 2x – 3
x2 – x – 6
x3 – 2x2 – 3x÷
b3 – 4b
b3 + b
b2 – b – 2
b4 – 1÷
x2 – 6x + 8
x2 – 4
–x3 + x
–2x2 + 4x + 16÷
y2 – y – 2
y2 + 3y – 4
–y + 2
y2 – 3y + 2÷
k2 + km – 2m2
2k + 2m
2x2 – 16x + 32
x3 – x
t – 2
3
x3 + x2
x – 3–
y2 – y – 2
y + 4–
Dividing Rational ExpressionsDividing Rational Expressions
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Independent Practice
Solution follows…
Topic
8.2.2
Divide and simplify each expression.
7. 8.
9. 10.
a3 – 4a
–a2 + 2a
a2 + a – 2
a2 – a – 2÷
b2 – 1
b2 – 2b – 3
b2 – 2b + 1
b2 – 4b + 3÷
(m – v)2
m2 – v2
m2 – 3mv + 2v2
(m – 2v)2÷
x2 – 3x + 2
x2 + x – 2
–x + 2
x2 – 3x – 10÷
a2 – a – 2
a – 1–
1
m – 2v
m + v
–x + 5
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Independent Practice
Solution follows…
Topic
8.2.2
Divide and simplify each expression.
11.
12.
13.
2x2 – 5x – 12
4x2 + 8x + 3
x2 – 16
2x2 + 7x + 3÷
x2 – 9
x2 + 2x – 8÷
y + 5
y2 – 4y – 5
y2 + 4y – 5
y + 1÷
1
y2 – 6y + 5÷
t2 – t – 6
t2 + 6t + 9÷ (t2 – 4) ÷
t + 2
t + 3
1
x – 2
x – 3
1
(t + 2)2
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Independent Practice
Solution follows…
Topic
8.2.2
Simplify these rational expressions.
14.
15.
16.
17.
k2 – 5k + 6
k2 + 2k – 8
–2k2 – 6k – 4
k2 – 2k – 3•
k2 + 3k + 2
k2 + 5k + 4÷
–2v2 + 4vw
3v2 – 4vw + w2×
v2 – w2
–2vw + 3w2÷
v3 – vw2
6v2 + 4vw2 – 2w3
–2v2 + 5vw – 3w2
–4v2 + 4vw + 8w2×
m2 + 2mn + n2
m2n – 3mn2÷ ÷ ÷
4m2 + 5mn + n2
2m2 – 5mn – 3n2
2m2 + 3mn + n2
–m2 + 3mn – 2n2
–m2 + 3mn – 2n2
–2m2n + 5mn2
m2 + 2mn + n2
m2n – 3mn2÷ ÷ ÷
4m2 + 5mn + n2
2m2 – 5mn – 3n2
2m2 + 3mn + n2
–m2 + 3mn – 2n2
–m2 + 3mn – 2n2
–2m2n + 5mn2
2m – 5n
4m + n–
–2
1
(m + n)2(2m + n)2(–2m + 5n)
(4m + n)(m – 2n)2(–m + n)2
Dividing Rational ExpressionsDividing Rational Expressions