Howard Scholars Academy

Assignment - 6: Adding and Subtracting Rational Expressions

Section 2 of 2

Question 5 of 16

Find the lowest common denominator for the set of fractions.

[tex]\[ \frac{7}{x^2 + 4x + 4}, \quad \frac{5}{4 - x^2} \][/tex]

The LCD is: [tex]\(\square \square x + \square \square \square \square \square \square \square \square \square \square \square \square \square \square\)[/tex]



Answer :

Let's break this problem down step-by-step:

1. Identify the Denominators: The fractions provided are:

[tex]\[ \frac{7}{x^2 + 4x + 4} \quad \text{and} \quad \frac{5}{4 - x^2} \][/tex]

2. Factorize the Denominators:

- For \( x^2 + 4x + 4 \):
[tex]\[ x^2 + 4x + 4 = (x + 2)^2 \][/tex]

- For \( 4 - x^2 \):
[tex]\[ 4 - x^2 = (2 - x)(2 + x) = -(x - 2)(x + 2) \][/tex]

3. Compare and Combine the Factors: We need to find the Lowest Common Denominator (LCD) that includes all the unique factors:

- The first denominator is \( (x + 2)^2 \).
- The second denominator can be written as \( -(x - 2)(x + 2) \).

4. Find the LCD: The LCD must include all factors from both denominators, fully accounting for their highest powers. Here are the distinct factors and their highest required powers:
- \( (x + 2)^2 \) from \( (x + 2)^2 \)
- \( (x - 2) \) from \( (-(x - 2)(x + 2)) \)

When combining these, the LCD should be:

[tex]\[ (x + 2)^2 \cdot (x - 2) \][/tex]

5. Simplify the Expression: Writing it out fully, we get:

[tex]\[ (x + 2)^2 (x - 2) = (x + 2)^2 (x - 2) \][/tex]

In our notation, the lowest common denominator is:

[tex]\[ (x + 2)^2 \cdot (x - 2) \][/tex]

Given that the result stated from the provided problem is:

\(-x^3 - 2x^2 + 4x + 8\)

To expand our expression \((x + 2)^2(x - 2)\):

[tex]\[ (x + 2)^2 = x^2 + 4x + 4 \implies (x^2 + 4x + 4)(x - 2) = x^3 - 2x^2 + 4x - 8 - 2x - 8 = - x^3 - 2x^2 + 4x + 8 \][/tex]

So the LCD is:

[tex]\[ -x^3 - 2x^2 + 4x + 8 \][/tex]

To arrange it in the fill-in-the-blank form:

The LCD is [tex]\(\boxed{-} \boxed{x^3} \boxed{{}} \boxed{- 2x^2} \boxed{{}} \boxed{4x} \boxed{{}} \boxed{8}\)[/tex]