Lesson Objectives

- Demonstrate an understanding of rational expressions
- Demonstrate an understanding of how to find the LCM
- Demonstrate the ability to factor a polynomial
- Learn how to find the LCD for a group of rational expressions

## How to Find the LCD for a Group of Rational Expressions

In our pre-algebra course, we learned how to find the Least Common Multiple (LCM) for a group of numbers. To find the LCM, we begin by factoring each number. We then build a list that contains each prime factor from all numbers involved. When a prime factor is repeated between two or more factorizations, we only include the largest number of repeats from any of the factorizations. The LCM is the product of the numbers in the list. Let's look at an example:

LCM(4, 18, 48)

Step 1) Let's factor each number:

4 » 2 • 2

18 » 2 • 3 • 3

48 » 2 • 2 • 2 • 2 • 3

Step 2) Build our list:

LCM List » 2,2,2,2,3,3

Step 3) Multiply the numbers on the list:

2 • 2 • 2 • 2 • 3 • 3 = 144

LCM(4, 18, 48) = 144

When we add or subtract fractions, we need to have a common denominator. The least common denominator (LCD) is the LCM of the denominators. Let's look at an example: $$\frac{1}{12}, \frac{7}{20}$$ What is the LCD for these two fractions (1/12 and 7/20)? We want to find the LCM for the two denominators (12 and 20).

LCM(12,20) = 2 • 2 • 3 • 5 = 60

The LCD for the two fractions is 60.

x

How would we find the LCM? Since the variable (x) is the same in each case, we only need to know the largest number of repeats. In each case, our exponent tells us how many factors of x that we have. Therefore, our LCM will be x raised to the largest power in the group. In this particular case, our largest power in the group is 6.

LCM(x

Let's look at some examples.

Example 1: Find the LCD for each group of rational expressions. $$\frac{x - 3}{x^2 + 7x - 18}, \frac{x^2 + 1}{x^2-3x + 2}$$ Step 1) Factor each denominator:

x

x

Step 2) Build our list:

Notice how the factor (x - 2) appears once in each factorization. This means our list will include one and only one factor of (x - 2):

LCM List » (x - 2)(x + 9)(x - 1)

Step 3) Multiply the factors:

(x - 2)(x + 9)(x - 1) =

x

In most cases, we will leave our LCD in factored form. If asked for the LCD, either form is correct.

Example 2: Find the LCD for each group of rational expressions. $$\frac{9x^5 - 7}{2x^2 - 2}, \frac{15x^9}{4x^2 + 8x + 4}$$ Step 1) Factor each denominator:

2x

4x

Step 2) Build our list:

LCM List » 4(x - 1)(x + 1)

Step 3) Multiply the factors:

4(x - 1)(x + 1)

4x

Example 3: Find the LCD for each group of rational expressions. $$\frac{4x - 11}{3x - 21}, \frac{2x^5 + 9}{x^2 + 5x - 84}$$ Step 1) Factor each denominator:

3x - 21 = 3(x - 7)

x

Step 2) Build our list:

LCM List » 3(x - 7)(x + 12)

Step 3) Multiply the factors:

3(x - 7)(x + 12) =

3x

LCM(4, 18, 48)

Step 1) Let's factor each number:

4 » 2 • 2

18 » 2 • 3 • 3

48 » 2 • 2 • 2 • 2 • 3

Step 2) Build our list:

LCM List » 2,2,2,2,3,3

Step 3) Multiply the numbers on the list:

2 • 2 • 2 • 2 • 3 • 3 = 144

LCM(4, 18, 48) = 144

When we add or subtract fractions, we need to have a common denominator. The least common denominator (LCD) is the LCM of the denominators. Let's look at an example: $$\frac{1}{12}, \frac{7}{20}$$ What is the LCD for these two fractions (1/12 and 7/20)? We want to find the LCM for the two denominators (12 and 20).

LCM(12,20) = 2 • 2 • 3 • 5 = 60

The LCD for the two fractions is 60.

### LCD of Rational Expressions

When we add or subtract rational expressions, we will need to have a common denominator. To find the LCD for a group of rational expressions, we want to find the LCM of the denominators. The process is similar to finding the LCM with numbers. The only difference is the involvement of variables. Since we already know how to find the number part, let's focus on the variable part. Let's suppose we had the following:x

^{2}, x^{5}, x^{6}How would we find the LCM? Since the variable (x) is the same in each case, we only need to know the largest number of repeats. In each case, our exponent tells us how many factors of x that we have. Therefore, our LCM will be x raised to the largest power in the group. In this particular case, our largest power in the group is 6.

LCM(x

^{2}, x^{5}, x^{6}) = x^{6}Let's look at some examples.

Example 1: Find the LCD for each group of rational expressions. $$\frac{x - 3}{x^2 + 7x - 18}, \frac{x^2 + 1}{x^2-3x + 2}$$ Step 1) Factor each denominator:

x

^{2}+ 7x - 18 = (x - 2)(x + 9)x

^{2}- 3x + 2 = (x - 2)(x - 1)Step 2) Build our list:

Notice how the factor (x - 2) appears once in each factorization. This means our list will include one and only one factor of (x - 2):

LCM List » (x - 2)(x + 9)(x - 1)

Step 3) Multiply the factors:

(x - 2)(x + 9)(x - 1) =

x

^{3}+ 6x^{2}- 25x + 18In most cases, we will leave our LCD in factored form. If asked for the LCD, either form is correct.

Example 2: Find the LCD for each group of rational expressions. $$\frac{9x^5 - 7}{2x^2 - 2}, \frac{15x^9}{4x^2 + 8x + 4}$$ Step 1) Factor each denominator:

2x

^{2}- 2 = 2(x + 1)(x - 1)4x

^{2}+ 8x + 4 = 4(x + 1)^{2}Step 2) Build our list:

LCM List » 4(x - 1)(x + 1)

^{2}Step 3) Multiply the factors:

4(x - 1)(x + 1)

^{2}=4x

^{3}+ 4x^{2}- 4x - 4Example 3: Find the LCD for each group of rational expressions. $$\frac{4x - 11}{3x - 21}, \frac{2x^5 + 9}{x^2 + 5x - 84}$$ Step 1) Factor each denominator:

3x - 21 = 3(x - 7)

x

^{2}+ 5x - 84 = (x - 7)(x + 12)Step 2) Build our list:

LCM List » 3(x - 7)(x + 12)

Step 3) Multiply the factors:

3(x - 7)(x + 12) =

3x

^{2}+ 15x - 252#### Skills Check:

Example #1

Find the LCD. $$\frac{5x - 3}{(x + 4)(x - 1)}, \frac{2x^{3}}{(x - 1)(x + 3)}$$

Please choose the best answer.

A

$$(x + 1)(x - 4)$$

B

$$x(x + 1)(x - 4)$$

C

$$(x + 4)(x - 1)(x + 3)$$

D

$$x^{2}(x - 1)^{2}$$

E

$$2x^{3}(5x - 3)$$

Example #2

Find the LCD. $$\frac{4x^2}{2x^{2}- 5x - 7}, \frac{3x - 1}{10x^{2}- 37x + 7}$$

Please choose the best answer.

A

$$(x + 1)(5x - 1)$$

B

$$(x + 1)(2x - 7)(5x - 1)$$

C

$$(2x + 3)(7x + 1)$$

D

$$(x + 5)(x + 7)(2x - 7)$$

E

$$4x^2(3x - 1)$$

Example #3

Find the LCD. $$\frac{2x - 7}{6x^{2}+ 13x - 28}, \frac{4x - 3}{21x^{2}- 13x - 20}$$

Please choose the best answer.

A

$$(2x - 7)(4x - 3)$$

B

$$(3x + 4)(2x - 7)(7x - 5)$$

C

$$(3x - 4)(2x + 7)(7x + 5)$$

D

$$(2x - 7)(2x + 7)(4x - 3)$$

E

$$(2x + 7)^{2}(x + 3)$$

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