8.6: Adding and Subtracting Rational Expressions
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Basic Rule
We are now in a position to study the process of adding and subtracting rational expressions. There is a most basic rule to which we must strictly adhere if we wish to conveniently add or subtract rational expressions.
To add or subtract rational expressions conveniently, they should have the same denominators.
Thus, to add or subtract two or more rational expressions conveniently, we must ensure that they all have the same denominator. The denominator that is most convenient is the LCD.
Fractions With The Same Denominator
To add (or subtract) two or more rational expressions with the same denominators, add (or subtract) the numerators and place the result over the LCD. Reduce if necessary. Symbolically,
ac+bc=a+bc
ac−bc=a−bc
Note that we combine only the numerators.
Sample Set A
Add or subtract the following rational expressions.
16+36 The denominators are the same. Add the numerators.16+36=1+36=46 Reduce 16+36=23
5x+8x The denominators are the same. Add the numerators 5x+5+8x=13x
2aby2w−5by2w The denominators are the same. Subtract the numerators 2aby2w−5by2w=2ab−5by2w
3x2+x+2x−7+x2−4x+1x−7 The denominators are the same. Add the numerators3x2+x+2x−7+x2−4x+1x−7=3x2+x+2+x2−4x+1x−7=4x2−3x+3x−7
5y+32y−5−2y+42y−5 The denominators are the same. Subtract the numerators. But be careful to subtract the entire numerator. Use parentheses!5y+32y−5−2y+42y−5=5y+3−(2y+4)2y−5=5y+3−2y−42y−5=3y−12y−5
Note: 5y+32y−5−2y+42y−5
The term −2y+42y−5 could be written as
+−(2y+4)2y−5=−2y−42y−5
A common mistake is to write:
−2y+42y−5 as −2y+42y−5
This is not correct, as the negative sign is not being applied to the entire numerator
3x2+4x+5(x+6)(x−2)+2x2+x+6x2+4x−12−x2−4x−6x2+4x−12
Factor the denominators to determine if they're the same:
3x2+4x+5(x+6)(x−2)+2x2+x+6(x+6)(x−2)−x2−4x−6(x+6)(x−2)
The denominators are the same. Combine the numerators being careful to note the negative sign.
3x2+4x+5+2x2+x+6−(x2−4x+6)(x+6)(x−2)
3x2+4x+5+2x2+x+6−x2+4x+6(x+6)(x−2)
4x2+9x+17(x+6)(x−2)
Practice Set A
Add or Subtract the following rational expressions.
49+29
- Answer
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23
3b+2b
- Answer
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5b
5x2y2−3x2y2
- Answer
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xy2
x+yx−y+2x+3yx−y
- Answer
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3x+4yx−y
4x2−x+43x+10−x2+2x+53x+10
- Answer
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3x2−3x−13x+10
x(x+1)x(2x+3)+3x2−x+72x2+3x
- Answer
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4x2+7x(2x+3)
4x+3x2−x−6−8x−4(x+2)(x−3)
- Answer
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−4x+7(x+2)(x−3)
5a2+a−42a(a−6)+2a2+3a+42a2−12a+a2+22a2−12a
- Answer
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4a2+2a+1a(a−6)
8x2+x−1x2−6x+8+2x2+3xx2−6x+8−5x2+3x−4(x−4)(x−2)
- Answer
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5x2+x+3(x−4)(x−2)
Fractions with Different Denominators
Sample Set B
Add or Subtract the following rational expressions.
4a3y+2a8y2.The denominators are not the same. Find the LCD. By inspection, the LCD is9y2The denominator of the first rational expression has been multiplied by 3y?9y2+2a9y2so the numerator must be multiplied by 3y4a⋅3y=12ay12ay9y2+2a9y2The denominators are now the same. Add the numerators.12ay+2a9y2
3bb+2+5bb−3 The denominators are not the same. The LCD is (b+2)(b−3)?(b+2)(b−3)+?(b+2)(b−3) The denominator of the first rational expression has been multiplied by b−3, so the numerator must be multiplied by b−3.3b(b−3)3b(b−3)(b+2)(b−3)+?(b+2)(b−3) The denominator of the second rational expression has been multiplied by b+2, so the numerator must be multiplied by b+2.5b(b+2).3b(b−3)(b+2)(b−3)+5b(b+2)(b+2)(b−3) The denominators are now the same. Add the numerators.
3b(b−3)+5b(b+2)(b−3)(b+2)=3b2−9b+5b2+10b(b−3)(b+2)=8b2+b(b−3)(b+2)
x+3x−1+x−24x+4 The denominators are not the same Find the LCDx+3x−1+x−24(x+1) The LCD is (x+1)(x−1)?4(x+1)(x−1)+?4(x+1)(x−1) The denominator of the first rational expression has been multiplied by 4(x+1) the numerator must be multiplied by 4(x+1)4(x+3)(x+1)4(x+3)(x+1)4(x+1)(x−1)+?4(x+1)(x−1) The denominator of the second rational expression has been multiplied by (x−1) so the numerator must be multiplied by x−1(x−1)(x−2)4(x+3)(x+1)4(x+1)(x−1)+(x−1)(x−2)4(x+1)(x−2) The denominator are now the same. Add the numerators4(x+3)(x+1)+(x−1)(x−2)4(x+1)(x−1)4(x2+4x+3)+x2−3x+24(x+1)(x−1)
4x2+16x+12+x2−3x+24(x+1)(x−1)=5x2+13x+144(x+1)(x−1)
x+5x2−7x+12+3x−1x2−2x−3Determine the LCDx+5(x−4)(x−3)+3x−1(x−3)(x+1)The LCD is (x−4)(x−3)(x+1)?(x−4)(x−3)(x+1)+?(x−4)(x−3)(x+1) The first numerator must be multipled by x+1 and the second by x−4(x+5)(x+1)(x−4)(x−3)(x+1)+(3x−1)(x−4)(x−4)(x−3)(x+1) The denominators are now the same. Add the numerators (x+5)(x+1)+(3x−1)(x−4)(x−4)(x−3)(x+1)x2+6x+5=3x2+−13x+4(x−4)(x−3)(x+1)4x2−7x+9(x−4)(x−3)(x+1)
a+4a2+5a+6−a−4a2−5a−24Determine the LCDa+4(a+3)(a+2)−a−4(a+3)(a−8) The LCD is (a+3)(a+2)(a−8)?(a+3)(a+2)(a−8)−?(a+3)(a+2)(a−8) The first numerator must be multipled by a−8 and the second by a+2.(a+4)(a−8)(a+3)(a+2)(a−8)−(a−4)(a+2)(a+3)(a+2)(a−8) The denominators are now the same. Subtract the numerators.(a+4)(a−8)−(a−4)(a+2)(a+3)(a+2)(a−8)a2−4a−32−(a2−2a−8)(a+3)(a+2)(a−8)a2−4a−32−a2+2a+8(a+3)(a+2)(a−8)−2a−24(a+3)(a+2)(a−8) Factor −2 from the numerator.−2(a+12)(a+3)(a+2)(a−8)
3x7−x+5xx−7 The denominators are nearly the same. They differ only in sign Our technique is to factor −1 from one of them3x7−x=3x−(x−7)=−3xx−73x7−x+5xx−7=−3xx−7+5xx−7=−3x+5xx−7=2xx−7
Practice Set B
Add or Subtract the following rational expressions.
3x4a2+5x12a3
- Answer
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9ax+5x12a3
5bb+1+3bb−2
- Answer
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8b2−7b(b+1)(b−2)
a−7a+2+a−2a+3
- Answer
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2a2−4a−25(a+2)(a+3)
4x+1x+3−x+5x−3
- Answer
-
3x2−19x−18(x+3)(x−3)
2y−3y+3y+1y+4
- Answer
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5y2+6y−12y(y+4)
a−7a2−3a+2+a+2a2−6a+8
- Answer
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2a2−10a+26(a−2)(a−1)(a−4)
6b2+6b+9−2b2+4b+4
- Answer
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4b2+12b+6(b+3)2(b+2)2
xx+4−x−23x−3
- Answer
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2x2−5x+83(x+4)(x−2)
5x4−x+7xx−4
- Answer
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2xx−4
Sample Set C
Combine the following rational expressions.
3+7x−1Rewrite the expression31+7x−1The LCD is x−13(x−1)x−1+7x−1=3x−3x−1+7x−1=3x−3+7x−1=3x+4x−1
3y+4−y2−y+3y−6Rewrite the expression.3y+41−y2−y+3y−6The LCD is y−6(3y+4)(y−6)y−6−y−y+3y−6=(3y+4)(y−6)−(y2−y+3)y−6=3y2−14y−24−y2+y−3y−6=2y2−13y−27y−6
Practice Set C
Simplify 8+3x−6
- Answer
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8x−45x−6
Simplify 2a−5−a2+2a−1a+3
- Answer
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a2−a−14a+3
Exercises
For the following problems, add or subtract the rational expressions.
38+18
- Answer
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12
19+49
710−25
- Answer
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310
34−512
34x+54x
- Answer
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2x
27y+37y
6y5x+8y5x
- Answer
-
14y5x
9a7b+3a7b
15n2m−6n2m
- Answer
-
9n2m
8p11q−3p11q
y+4y−6+y+8y−6
- Answer
-
2y+12y−6
y−1y+4+y+7y+4
a+6a−1+3a+5a−1
- Answer
-
4a+11a−1
5a+1a+7+2a−6a+7
x+15x+x+35x
- Answer
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2x+45x
a−6a+2+a−2a+2
b+1b−3+b+2b−3
- Answer
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2b+3b−3
a+2a−5−a+3a−5
b+7b−6−b−1b−6
- Answer
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8b−6
2b+3b+1−b−4b+1
3y+4y+8−2y−5y+8
- Answer
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y+9y+8
2a−7a−9+3a+5a−9
8x−1x+2−15x+7x+2
- Answer
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−7x−8x+2
72x2+16x3
23x+46x2
- Answer
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2(x+1)3x2
56y3−218y5
25a2−110a3
- Answer
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4a−110a3
3x+1+5x−2
4x−6+1x−1
- Answer
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5(x−2)(x−6)(x−1)
2aa+1−3aa+4
6yy+4+2yy+3
- Answer
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2y(4y+13)(y+4)(y+3)
x−1x−3+x+4x−4
x+2x−5+x−1x+2
- Answer
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2x2−2x+9(x−5)(x+2)
a+3a−3−a+2a−2
y+1y−1−y+4y−4
- Answer
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−6y(y−1)(y−4)
x−1(x+2)(x−3)+x+4x−3
y+2(y+1)(y+6)+y−2y+6
- Answer
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y2(y+1)(y+6)
2a+1(a+3)(a−3)−a+2a+3
3a+5(a+4)(a−1)−2a−1a−1
- Answer
-
−2a2−4a+9(a+4)(a−1)
2xx2−3x+2+3x−2
4aa2−2a−3+3a+1
- Answer
-
7a−9(a+1)(a−3)
3yy2−7y+12−y2y−3
x−1x2+6x+8+x+3x2+2x−8
- Answer
-
2(x2+x+4)(x+2)(x−2)(x+4)
a−4a2+2a−3+a+2a2+3a−4
x−1x2+6x+8+x+3x2+2x−8
- Answer
-
2(x2+x+4)(x+2)(x−2)(x+4)
a−4a2+2a−3+a+2a2+3a−4
b−3b2+9b+20+b+4b2+b−12
- Answer
-
2b2+3b+29(b−3)(b+4)(b+5)
y−1y2+4y−12−y+3y2+6y−16
x+3x2+9x+13−x−5x2−4
- Answer
-
−x+29(x−2)(x+2)(x+7)
x−1x2−4x+3+x+3x2−5x+6+2xx2−3x+2
4xx2+6x+8+3x2+x−6+x−1x2+x−12
- Answer
-
5x4−3x3−34x2+34x−60(x−2)(x+2)(x−3)(x+3)(x+4)
y+2y2−1+y−3y2−3y−4−y+3y2−5y+4
a−2a2−9a+18+a−2a2−4a−12−a−2a2−a−6
- Answer
-
(a+5)(a−2)(a+2)(a−3)(a−6)
y−2y2+6y+y+4y2+5y−6
a+1a3+3a2−a+6a2−a
- Answer
-
−a3−8a2−18a−1a2(a+3)(a−1)
43b2−12b−26b2−6b
32x5−4x4+−28x3+24x2
- Answer
-
−x3+2x2+6x+184x4(x−2)(x+3)
x+212x3+x+14x2+8x−12−x+316x2−32x+16
2xx2−9−x+14x2−12x−x−48x3
- Answer
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14x4−9x3−2x2+9x−368x3(x+3)(x−3)
4+3x+2
8+2x+6
- Answer
-
8x+50x+6
1+4x−7
3+5x−6
- Answer
-
3x−13x−6
−2+4xx+5
−1+3aa−1
- Answer
-
2a+1a−1
6−4yy+2
2x+x2−4x+1
- Answer
-
3x2+2x−4x+1
−3y+4y2+2y−5y+3
x+2+x2+4x−1
- Answer
-
2x2+x+2x−1
b+6+2b+5b−2
3x−1x−4−8
- Answer
-
−5x+31x−4
4y+5y+1−9
2y2+11y−1y+4−3y
- Answer
-
−(y2+y+1)y+4
5y2−2y+1y2+y−6−2
4a3+2a2+a−1a2+11a+28+3a
- Answer
-
7a3+35a2+85a−1(a+7)(a+4)
2x1−x+6xx−1
5m6−m+3mm−6
- Answer
-
−2mm−6
−a+78−3a+2a+13a−8
Exercises For Review
Simplify (x3y2z5)6(x2yz)2
- Answer
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x22y14z32
Write 6a−3b4c−2a−1b−5c3 so that only positive exponents appear.
Construct the graph of y=−2x+4
- Answer
-
Find the product x2−3x−4x2+6x+5⋅x2+5x+6x2−2x−8
Replace N with the proper quantity: x+3x−5=Nx2−7x+10
- Answer
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(x+3)(x−2)