To find the Lowest Common Denominator (LCD) of the given rational equation:
[tex]\[
\frac{-3}{x+2} + \frac{5x}{x-1} = \frac{x+2}{x^2 - 3x + 2}
\][/tex]
We'll follow these steps:
1. Identify the denominators:
- The first term has the denominator [tex]\(x+2\)[/tex].
- The second term has the denominator [tex]\(x-1\)[/tex].
- The third term has the denominator [tex]\(x^2 - 3x + 2\)[/tex].
2. Factorize the polynomial denominator to further simplify the equation:
- The denominator [tex]\(x^2 - 3x + 2\)[/tex] can be factored. We look for two numbers that multiply to [tex]\(2\)[/tex] (the constant term) and add to [tex]\(-3\)[/tex] (the coefficient of the [tex]\(x\)[/tex] term):
[tex]\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\][/tex]
3. List all distinct linear factors that appear in any denominator:
- From the first term: [tex]\(x+2\)[/tex]
- From the second term: [tex]\(x-1\)[/tex]
- From the third term (factored form): [tex]\((x-1)(x-2)\)[/tex]
Combining all distinct factors from the above, we get [tex]\((x+2)\)[/tex], [tex]\((x-1)\)[/tex], and [tex]\((x-2)\)[/tex].
4. Combine all factors to form the LCD:
- The LCD must include each distinct factor to the highest power it appears in any denominator.
Thus, the LCD for the given rational equation is:
[tex]\[
(x+2)(x-1)(x-2)
\][/tex]
Looking at the provided answer choices:
A. [tex]\((x+2)(x-1)(x^2 - 3x + 2)\)[/tex]
B. [tex]\((x+2)(x-1)(x-2)\)[/tex]
C. [tex]\((x+2)(x-1)\)[/tex]
D. [tex]\(5x(x+2)\)[/tex]
The correct answer is:
B. [tex]\((x+2)(x-1)(x-2)\)[/tex]
