complex fractions. objective simplify complex fractions lets review fraction rules first…………
TRANSCRIPT
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Complex fractions
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Objective
• Simplify complex fractions
• Lets Review fraction rules first…………..
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Multiplying Fractions
0d and 0b,db
ca
d
c
b
a
Multiply . - 521
·34
- 521
·34
=- 521
·34
1
7
- 57
·14
= =- 528
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Multiplying Rational Expressions
1. Factor all numerators and denominators completely.
2. Divide out common factors.3. Multiply numerators together and
multiply denominators together.
Multiply .yx
22z
11z
y18x-52
32
52
32
yx
22z
11z
y18x-4
2
y
36z
52
32
yx
22z
11z
y18x-
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Dividing Two Fractions
0c and 0d , 0b ,bc
ad
c
d
b
a
d
c
b
a
Divide . - 2 9
59
=- 2 9
59
- 2 9
·95
1- 2 5
=
1
- 2 9
·95
=
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Dividing Rational Expressions
Invert the divisor (the second fraction) and multiply
Divide .3017-x
1
127xx
122
1
3017-x
187xx
1
3017-x
1
187xx
1 2
222
1
15)2)(x(x
2)9)(x(x
1
9)(x
15)(x
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Adding/Subtracting Fractions
0c ,c
ba
c
b
c
a 0c ,
c
ba
c
b
c
a
712
= 512
212
+
Add . 512
212
+
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Common Denominators
1. Add or subtract the numerators.2. Place the sum or difference of the
numerators found in step 1 over the common denominator.
3. Simplify the fraction if possible.
Subtract .5
6
5
7-2x
5
13-2x
5
6-7-2x
5
6
5
7-2x
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Common Denominators
a.) Add .12ww
4-2w-
12ww
53w22
Example:
12ww
4-2w-53w
12ww
4-2w-
12ww
53w222
1)(w
1
1)(w
1w2
12ww
4-2w-53w2
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Common Denominators
b.) Subtract
.649x
29x-x
649x
54x2
2
2
2
649x
29)x-(x-54x
649x
29x-x
649x
54x2
22
2
2
2
2
649x
24x3x
649x
29xx-54x2
2
2
22
8)(3x
3)(x
8)8)(3x(3x
8)3)(3x(x
Example:
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Unlike Denominators
1. Determine the LCD.2. Rewrite each fraction as an
equivalent fraction with the LCD.3. Add or subtract the numerators
while maintaining the LCD.4. When possible, factor the
remaining numerator and simplify the fraction.
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Unlike Denominators
a.)w
5
2w
3
2w
2w
w
5
w
w
2w
3
The LCD is w(w+2).
2)w(w
2)5(w
2)w(w
3w
2)w(w
105w
2)w(w
3w
2)w(w
108w
answers. acceptable also are and 2ww
108w
2)w(w
5)2(4w2
Example:
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Unlike Denominators
b.)3x
1
4-4x
x The LCD is 12x(x – 1).
3x
1
1)-4(x
x
1)-4(x
1)-4(x
3x
1
3x
3x
1)-4(x
x
1)-12x(x
44x3x
1)-12x(x
1)-4(x
1)-12x(x
3x 22
This cannot be factored any further.
Example:
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Complex Fractions
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Simplifying Complex Fractions
A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator.
454
3xx
3x
ba9-a
ba
Example:
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So how can we simplify them?
• Remember, fractions are just division problems.• We can rewrite the complex fraction as a division
problem with two fractions.• This division problem then changes to multiplication
by the reciprocal.
5
62
3
5
6
2
3
5
6
3
2
5
4
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Simplifying Complex Fractions Rule
• Any complex fraction
dcba
Where b ≠ 0, c ≠ 0, and d ≠ 0, may be expressed as:
bc
ad
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What if we have mixed numbers in the complex fraction?
• If we have mixed numbers, we treat it as an addition problem with unlike denominators.
• We want to be working with two fractions, so make sure the numerator is one fraction, and the denominator is one fraction
• Now we can rewrite the complex fraction as a division of two fractions
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Example
21
25
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Try on your own…
4
11
3
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What about complex rational expression?
• Treat the complex rational expression as a division problem
• Add any rational expressions to form rational expressions in the numerator and denominator
• Factor• Simplify• “Bad” values
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Ex. 2: Simplify .11
11
yx
yx
xyx
xyy
xyx
xyy
yx
yx
11
11
xyxy
xyxy
← The LCD is xy for both the numerator and the denominator.
← Add to simplify the numerator and subtract to simplify the denominator.
xy
xy
xy
xy
← Multiply the numerator by the reciprocal of the
denominator.
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Ex. 2: Simplify .11
11
yx
yx
xy
xy
xy
xy
← Eliminate common factors.
xy
xy
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Example
x 1
xx 1
(x 1
x) (x 1)
(x 2 1
x)
1
x 1
(x 1)(x 1)
x
1
x 1
x 1
x, x 0, 1
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Example
x 2
1 5
x 6
x 2
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Try on your own
2
3x1
x
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One more for you
x 16
xx 2 8x 16
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Ex. 3: Simplify
348
11
41
4
xx
xx
348)3)(11(
41)4)(4(
xxxxxx ← The LCD of the numerator is x +
4, and the LCD of the denominator is x – 3.
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Ex. 3: Simplify
348
11
41
4
xx
xx
348338
41168
2
2
xxxxxx
← FOIL the top and don’t forget to subtract the 1 and add the 48 on the bottom.
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Ex. 3: Simplify
348
11
41
4
xx
xx
3158
4158
2
2
xxx
xxx
← Simplify by subtracting the 1 in the numerator and adding the 48 in the denominator.
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Ex. 3: Simplify
348
11
41
4
xx
xx
158
3
4
1582
2
xx
x
x
xx
← Multiply by the reciprocal.
x2 + 8x +15 is a common factor that can be eliminated.
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Ex. 3: Simplify
348
11
41
4
xx
xx
4
3
x
x ← Simplify
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Model Problems5
31)1
2
x
x
12)
1y
yy
2433)
363
x
x
4)1 1
x yx
x y
2 65)
2 3
k k
k k
71
26)
31
2
y
y
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Homework
• Practice Sheet