55 addition and subtraction of rational expressions
TRANSCRIPT
Addition and Subtraction of Rational Expressions
Frank Ma © 2011
Addition and Subtraction of Rational ExpressionsOnly fractions with the same denominator may be added or subtracted directly.
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer.
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ =
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128
32
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128 = 3
232
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128 = 3
232
b. 3x2x – 3 – 6 – x
2x – 3
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128 = 3
232
b. 3x2x – 3 – 6 – x
2x – 3 = 3x – (6 – x) 2x – 3
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128 = 3
232
b. 3x2x – 3 – 6 – x
2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x
2x – 3
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128 = 3
232
b. 3x2x – 3 – 6 – x
2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x
2x – 3
= 4x – 6 2x – 3
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128 = 3
232
b. 3x2x – 3 – 6 – x
2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x
2x – 3
= 4x – 6 2x – 3 = 2(2x – 3)
2x – 3
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128 = 3
232
b. 3x2x – 3 – 6 – x
2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x
2x – 3
= 4x – 6 2x – 3 = 2(2x – 3)
2x – 3
Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or subtracted directly.
A BD D
± = A±BD
Usually we put the result in the factored form, cancel the common factor and give the simplified answer. Example A. Add and subtract and simplify the answer.
a. 5 78 8+ = 5 + 7
8 = 128 = 3
232
b. 3x2x – 3 – 6 – x
2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x
2x – 3
= 4x – 6 2x – 3 = 2(2x – 3)
2x – 3 = 2
To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator.
Addition and Subtraction of Rational Expressions
To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator. The easiest common denominator to work with is their LCM.
Addition and Subtraction of Rational Expressions
To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator. The easiest common denominator to work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier MethodGiven the fraction , to convert it into denominator D,
the new numerator is
AB
AB
*
D.
To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator. The easiest common denominator to work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier MethodGiven the fraction , to convert it into denominator D,
the new numerator is
Example B. a. Convert to a fraction with denominator 12.
AB
AB
*
D.
54
In practice, we write this as AB = A
B * D D
new numerator
To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator. The easiest common denominator to work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier MethodGiven the fraction , to convert it into denominator D,
the new numerator is
Example B. a. Convert to a fraction with denominator 12.
AB
AB
*
D.
54
In practice, we write this as AB = A
B * D D
54 =
new numerator
To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator. The easiest common denominator to work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier MethodGiven the fraction , to convert it into denominator D,
the new numerator is
Example B. a. Convert to a fraction with denominator 12.
AB
AB
*
D.
54
54 * 12
In practice, we write this as AB = A
B * D D
54 = 12
new numerator
new numerator
To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator. The easiest common denominator to work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier MethodGiven the fraction , to convert it into denominator D,
the new numerator is
Example B. a. Convert to a fraction with denominator 12.
AB
AB
*
D.
54
54 * 12
3
In practice, we write this as AB = A
B * D D
54 = 12
new numerator
new numerator
To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator. The easiest common denominator to work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method Given the fraction , to convert it into denominator D,
the new numerator is
Example B. a. Convert to a fraction with denominator 12.
AB
AB
*
D.
54
54 * 12
3 1512
In practice, we write this as AB = A
B * D D
54 = 12 =
new numerator
b. Convert into an expression with denominator 12xy2.
Addition and Subtraction of Rational Expressions3x4y
Addition and Subtraction of Rational Expressions3x4y
3x4y
b. Convert into an expression with denominator 12xy2.
Addition and Subtraction of Rational Expressions3x4y
*12xy23x4y = 3x
4y 12xy2
new numerator
b. Convert into an expression with denominator 12xy2.
Addition and Subtraction of Rational Expressions3x4y
*12xy23x4y = 3x
4y 12xy23xy
b. Convert into an expression with denominator 12xy2.
Addition and Subtraction of Rational Expressions3x4y
*12xy23x4y = 3x
4y 12xy2 = 9x2y12xy2
3xy
b. Convert into an expression with denominator 12xy2.
Addition and Subtraction of Rational Expressions3x4y
*12xy2
x + 12x + 3
3x4y = 3x
4y 12xy2 = 9x2y12xy2
3xy
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
Addition and Subtraction of Rational Expressions3x4y
*12xy2
x + 12x + 3
x + 12x + 3
3x4y = 3x
4y 12xy2 = 9x2y12xy2
3xy
= x + 12x + 3 * (4x2 – 9) (4x2 – 9)
new numerator
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
Addition and Subtraction of Rational Expressions3x4y
*12xy2
x + 12x + 3
x + 12x + 3
3x4y = 3x
4y 12xy2 = 9x2y12xy2
3xy
= x + 12x + 3 * (4x2 – 9) (4x2 – 9)
= x + 12x + 3 * (2x + 3)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
Addition and Subtraction of Rational Expressions3x4y
*12xy2
x + 12x + 3
x + 12x + 3
3x4y = 3x
4y 12xy2 = 9x2y12xy2
3xy
= x + 12x + 3 * (4x2 – 9) (4x2 – 9)
= x + 12x + 3 * (2x + 3)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
Addition and Subtraction of Rational Expressions3x4y
*12xy2
x + 12x + 3
x + 12x + 3
3x4y = 3x
4y 12xy2 = 9x2y12xy2
3xy
= x + 12x + 3 * (4x2 – 9) (4x2 – 9)
= x + 12x + 3 * (2x + 3)(2x – 3) (4x2 – 9)
= (x + 1)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
Addition and Subtraction of Rational Expressions3x4y
*12xy2
c. Convert into an expression denominator 4x2 – 9.x + 12x + 3
x + 12x + 3
3x4y = 3x
4y 12xy2 = 9x2y12xy2
3xy
= x + 12x + 3 * (4x2 – 9) (4x2 – 9)
= x + 12x + 3 * (2x + 3)(2x – 3) (4x2 – 9)
= (x + 1)(2x – 3) (4x2 – 9)
= 2x2 – x – 34x2 – 9
b. Convert into an expression with denominator 12xy2.
Addition and Subtraction of Rational ExpressionsWe give two methods of combining rational expressions below.
Addition and Subtraction of Rational ExpressionsWe give two methods of combining rational expressions below. The first one is the traditional method.
Addition and Subtraction of Rational ExpressionsWe give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result.
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example C. Combine
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2
3xy – x2y2
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2
3xy – x2y2
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2
3xy – x2y2
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2
3xy – x2y2
23xy = 6xy2
x2y2 = 3x2
6xy24y
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2
3xy – x2y2
23xy = 6xy2
x2y2 = 3x2
6xy24y
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2
3xy – x2y2
23xy = 6xy2
x2y2 = 3x2
6xy2
23xy – x
2y2 = 4y6xy2 – 3x2
6xy2Hence
4y
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2
3xy – x2y2
23xy = 6xy2
x2y2 = 3x2
6xy2
23xy – x
2y2 = 4y6xy2 – 3x2
6xy2 =Hence 4y – 3x2
6xy2
4y
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2
3xy – x2y2
23xy = 6xy2
x2y2 = 3x2
6xy2
23xy – x
2y2 = 4y6xy2 – 3x2
6xy2 =Hence 4y – 3x2
6xy2
This is simplified because the numerator is not factorable.
4y
We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2x2 + x – 2 =
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 =
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1)
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1)
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) = x2 + xLCD LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) = x2 + xLCD LCD
x – 1 2x2 + x – 1 =
x – 1 (2x – 1)(x + 1)
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) = x2 + xLCD LCD
x – 1 2x2 + x – 1 =
x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) = x2 + xLCD LCD
x – 1 2x2 + x – 1 =
x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) = x2 + xLCD LCD
x – 1 2x2 + x – 1 =
x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
= 2(x – 1) LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) = x2 + xLCD LCD
x – 1 2x2 + x – 1 =
x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
= 2(x – 1) = 2x – 2 LCD LCD
Example D. Combine
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD
x4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) = x2 + xLCD LCD
x – 1 2x2 + x – 1 =
x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
= 2(x – 1) = 2x – 2 LCD LCD
Hence x4x – 2 – x – 1
2x2 + x – 1 =x2 + xLCD – 2x – 2
LCD
Addition and Subtraction of Rational Expressionsx
4x – 2 – x – 1 2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
= x2 + x – 2x + 2LCD
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
= x2 + x – 2x + 2LCD
= x2 – x + 22(2x – 1)(x + 1)
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
= x2 + x – 2x + 2LCD
= x2 – x + 22(2x – 1)(x + 1)
This is simplified because the numerator is not factorable.
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
= x2 + x – 2x + 2LCD
= x2 – x + 22(2x – 1)(x + 1)
This is simplified because the numerator is not factorable. Multiplier Method
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
= x2 + x – 2x + 2LCD
= x2 – x + 22(2x – 1)(x + 1)
This is simplified because the numerator is not factorable. Multiplier Method The Multiplier Method is the same method used for fractional numbers.
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
= x2 + x – 2x + 2LCD
= x2 – x + 22(2x – 1)(x + 1)
This is simplified because the numerator is not factorable. Multiplier Method The Multiplier Method is the same method used for fractional numbers. We used the fact that x = (x * LCD) / LCD = x * 1
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
= x2 + x – 2x + 2LCD
= x2 – x + 22(2x – 1)(x + 1)
This is simplified because the numerator is not factorable. Multiplier Method The Multiplier Method is the same method used for fractional numbers. We used the fact that x = (x * LCD) / LCD = x * 1 then compute (x * LCD) using the Distributive Law.
Addition and Subtraction of Rational Expressions
= x2 + x – (2x – 2) LCD
x4x – 2 – x – 1
2x2 + x – 1
= x2 + xLCD – 2x – 2
LCD
= x2 + x – 2x + 2LCD
= x2 – x + 22(2x – 1)(x + 1)
This is simplified because the numerator is not factorable. Multiplier Method The Multiplier Method is the same method used for fractional numbers. We used the fact that x = (x * LCD) / LCD = x * 1 then compute (x * LCD) using the Distributive Law. Following is an example using this method.
Example E. Calculate
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
Example E. Calculate
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72.
Example E. Calculate
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Example E. Calculate
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )
Example E. Calculate
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72
Example E. Calculate
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication
Example E. Calculate
6
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication
Example E. Calculate
6 9
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example F. Combine
34xy2
– 5x6y
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example F. Combine
34xy2
– 5x6y
The LCD is 12 xy2.
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example F. Combine
34xy2
– 5x6y
The LCD is 12 xy2. Multiply then divide the problem by the LCD.
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example F. Combine
34xy2
– 5x6y
The LCD is 12 xy2. Multiply then divide the problem by the LCD. 3
4xy2 – 5x
6y ( ) * 12xy2 / (12xy2)
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example F. Combine
34xy2
– 5x6y
The LCD is 12 xy2. Multiply then divide the problem by the LCD. 3
4xy2 – 5x
6y ( ) * 12xy2 / (12xy2) Distribute
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example F. Combine
34xy2
– 5x6y
The LCD is 12 xy2. Multiply then divide the problem by the LCD. 3
4xy2 – 5x
6y ( ) * 12xy2 / (12xy2) Distribute3
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example F. Combine
34xy2
– 5x6y
The LCD is 12 xy2. Multiply then divide the problem by the LCD. 3
4xy2 – 5x
6y ( ) * 12xy2 / (12xy2) Distribute3 2xy
Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions712+ 5
8 – 49
The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD. 712+ 5
8 – 49( )* 72 72 Distribute the
multiplication= ( 42 + 45 – 32 )
72 5572=
Example F. Combine
34xy2
– 5x6y
The LCD is 12 xy2. Multiply then divide the problem by the LCD. 3
4xy2 – 5x
6y ( ) * 12xy2 / (12xy2) Distribute3 2xy
9 – 10x2y12xy2=
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4).
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence 5
x– 2 – 3
x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence 5
x– 2 – 3
x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)(x + 4)
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence 5
x– 2 – 3
x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)(x + 4) (x – 2)
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5x– 2
– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5x– 2
– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26(x – 2)(x + 4)=
2(x + 13)(x – 2)(x + 4)or
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5x– 2
– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26(x – 2)(x + 4)=
Example H. Combine
xx2 – 2x
– x – 1 x2 – 4
2(x + 13)(x – 2)(x + 4)or
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5x– 2
– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26(x – 2)(x + 4)=
Example H. Combine
xx2 – 2x
– x – 1 x2 – 4
Factor each denominator to find the LCD.
2(x + 13)(x – 2)(x + 4)or
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5x– 2
– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26(x – 2)(x + 4)=
Example H. Combine
xx2 – 2x
– x – 1 x2 – 4
Factor each denominator to find the LCD. x2 – 2x = x(x – 2)
2(x + 13)(x – 2)(x + 4)or
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5x– 2
– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26(x – 2)(x + 4)=
Example H. Combine
xx2 – 2x
– x – 1 x2 – 4
Factor each denominator to find the LCD. x2 – 2x = x(x – 2)x2 – 4 = (x – 2)(x + 2)
2(x + 13)(x – 2)(x + 4)or
Addition and Subtraction of Rational ExpressionsExample G. Combine
5x– 2
– 3x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5x– 2
– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26(x – 2)(x + 4)=
Example H. Combine
xx2 – 2x
– x – 1 x2 – 4
Factor each denominator to find the LCD. x2 – 2x = x(x – 2)x2 – 4 = (x – 2)(x + 2)Hence the LCD = x(x – 2)(x + 2).
2(x + 13)(x – 2)(x + 4)or
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
xx(x – 2)
– (x – 1) (x – 2)(x + 2)
=
xx2 – 2x
– x – 1 x2 – 4
* x( x – 2)(x + 2)
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
xx(x – 2)
– (x – 1) (x – 2)(x + 2)
[ ] LCD=
xx2 – 2x
– x – 1 x2 – 4
* x( x – 2)(x + 2)(x + 2)
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
xx(x – 2)
– (x – 1) (x – 2)(x + 2)
[ ] LCD=
xx2 – 2x
– x – 1 x2 – 4
* x( x – 2)(x + 2)(x + 2) x
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
xx(x – 2)
– (x – 1) (x – 2)(x + 2)
[ ] LCD=
xx2 – 2x
– x – 1 x2 – 4
* x( x – 2)(x + 2)(x + 2) x
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
xx(x – 2)
– (x – 1) (x – 2)(x + 2)
[ ] LCD=
xx2 – 2x
– x – 1 x2 – 4
=
[x(x + 2) – x(x – 1)] LCD
* x( x – 2)(x + 2)(x + 2) x
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
xx(x – 2)
– (x – 1) (x – 2)(x + 2)
[ ] LCD=
xx2 – 2x
– x – 1 x2 – 4
=
[x(x + 2) – x(x – 1)] LCD
=
[x2 + 2x – x2 + x)] LCD
* x( x – 2)(x + 2)(x + 2) x
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
xx(x – 2)
– (x – 1) (x – 2)(x + 2)
[ ] LCD=
xx2 – 2x
– x – 1 x2 – 4
=
[x(x + 2) – x(x – 1)] LCD
=
[x2 + 2x – x2 + x)] LCD
=
3xx (x – 2)(x + 2)
* x( x – 2)(x + 2)(x + 2) x
Multiply the problem by the LCD then put the result over the new denominator, the LCD.
Addition and Subtraction of Rational Expressions
xx(x – 2)
– (x – 1) (x – 2)(x + 2)
[ ] LCD=
xx2 – 2x
– x – 1 x2 – 4
=
[x(x + 2) – x(x – 1)] LCD
=
[x2 + 2x – x2 + x)] LCD
=
3xx (x – 2)(x + 2)=
3(x – 2)(x + 2)
Ex. A. Combine and simplify the answers. Addition and Subtraction of Rational Expressions
xx – 2
– 2x – 2 1. 2x
x – 2 + 4 x – 22.
3xx + 3
+ 6x + 3 3. – 2x
x – 4 + 8x – 44.
x + 22x – 1
– 2x – 1 5. 2x + 5
x – 2 – 4 – 3x2 – x6.
x2 – 2x – 2
– xx – 2 7. 9x2
3x – 2 – 43x – 28.
Ex. B. Combine and simplify the answers. 312 + 5
6 – 239. 11
12 + 58 – 7
610. –56 + 3
8 – 3 11.
12. 65xy2
– x6y 13.
34xy2
– 5x6y 15. 7
12xy – 5x
8y3 16.
54xy
– 7x6y2 14.
34xy2
– 5y12x2 17.
–56 – 7
12 + 2
+ 1 – 7x9y2
4 – 3x
Ex. C. Combine and simplify the answers. Addition and Subtraction of Rational Expressions
x2x – 4
– 23x – 6 18. 2x
3x + 9 – 42x + 619
. –32x + 1
+ 2x4x + 2 20. 2x – 3
x – 2 – 3x + 45 – 10x21.
3x + 16x – 4
– 2x + 32 – 3x22. –5x + 7
3x – 12 + 4x – 3–2x + 823.
xx – 2
– 2x – 3 24. 2x
3x + 1 + 4x – 625.
–32x + 1
+ 2x3x + 2 26. 2x – 3
x – 2 + 3x + 4x – 527.
3x + 1 + x + 3x2 – 428. x2 – 4x + 4
x – 4 – x + 5x2 – x – 229. x2 – 5x + 6
3x + 1 + 2x + 39 – x230. x2 – x – 6
3x – 4 – 2x + 5x2 + x – 631. x2 + 5x + 6
3x + 4 + 2x – 3–x2 – 2x + 332. x2 – x
5x – 4 – 3x – 51 – x233. x2 + 2x – 3