How To Find The Lcd Of Rational Expressions

Lesson Objectives

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

How to Find the LCD for a Grouping of Rational Expressions

In our pre-algebra course, we learned how to detect the Least Mutual Multiple (LCM) for a group of numbers. To discover the LCM, nosotros brainstorm by factoring each number. We then build a list that contains each prime number factor from all numbers involved. When a prime number factor is repeated betwixt ii or more factorizations, nosotros only include the largest number of repeats from whatsoever of the factorizations. The LCM is the product of the numbers in the list. Let'southward wait at an example:
LCM(4, eighteen, 48)
Step 1) Let's cistron each number:
iv » 2 • 2
18 » 2 • 3 • iii
48 » 2 • 2 • 2 • 2 • 3
Step 2) Build our list:
LCM List » two,2,two,2,3,3
Step 3) Multiply the numbers on the list:
2 • ii • 2 • ii • 3 • iii = 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 wait at an instance: $$\frac{one}{12}, \frac{7}{20}$$ What is the LCD for these two fractions (1/12 and 7/xx)? We desire to find the LCM for the two denominators (12 and 20).
LCM(12,xx) = 2 • 2 • iii • 5 = sixty
The LCD for the ii fractions is lx.

LCD of Rational Expressions

When we add together or subtract rational expressions, nosotros will need to have a common denominator. To observe 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 just difference is the involvement of variables. Since we already know how to find the number part, let'southward focus on the variable role. Let's suppose we had the following:
10², 10^five, x⁶
How would nosotros notice the LCM? Since the variable (10) is the same in each case, nosotros only demand to know the largest number of repeats. In each case, our exponent tells us how many factors of 10 that we have. Therefore, our LCM will be x raised to the largest power in the group. In this particular instance, our largest power in the group is 6.
LCM(x², x⁵, x⁶) = x⁶
Permit's look at some examples.
Example 1: Notice the LCD for each grouping of rational expressions. $$\frac{x - 3}{x^2 + 7x - eighteen}, \frac{x^2 + one}{10^2-3x + ii}$$ Step ane) Factor each denominator:
x^two + 7x - 18 = (ten - 2)(10 + 9)
x² - 3x + 2 = (x - ii)(x - 1)
Stride 2) Build our list:
Notice how the factor (ten - 2) appears one time in each factorization. This means our list will include one and only one cistron of (x - ii):
LCM List » (x - 2)(x + 9)(10 - 1)
Step iii) Multiply the factors:
(ten - 2)(x + ix)(x - 1) =
x³ + 6x² - 25x + 18
In almost cases, we will go out our LCD in factored form. If asked for the LCD, either form is correct.
Example two: Notice the LCD for each group of rational expressions. $$\frac{9x^v - 7}{2x^2 - ii}, \frac{15x^9}{4x^2 + 8x + four}$$ Footstep 1) Factor each denominator:
2x² - 2 = 2(ten + one)(x - ane)
4x² + 8x + 4 = 4(x + ane)²
Step 2) Build our list:
LCM List » iv(x - 1)(x + i)ⁱⁱ
Step 3) Multiply the factors:
4(10 - i)(x + 1)^two =
4xⁱⁱⁱ + 4x² - 4x - 4
Case 3: Observe the LCD for each group of rational expressions. $$\frac{4x - eleven}{3x - 21}, \frac{2x^5 + 9}{x^2 + 5x - 84}$$ Stride 1) Factor each denominator:
3x - 21 = iii(x - 7)
x² + 5x - 84 = (ten - 7)(x + 12)
Stride 2) Build our list:
LCM List » 3(10 - vii)(ten + 12)
Step 3) Multiply the factors:
3(x - 7)(x + 12) =
3x² + 15x - 252