adding and subtracting rational expressions
find the lcm for each group of expressions.
1. $2x^2 - 8x + 8$ and $3x^2 + 27x - 30$
$2x^2 - 8x + 8 = 2(x^2 - 4x + 4) = 2(x - 2)^2$
$3x^2 + 27x - 30 = 3(x^2 + 9x - 10) = 3(x + 10)(x - 1)$
$2 \times 3 \times (x - 2)^2 \times (x + 10) \times (x - 1)$
the lcm = $2 \times 3 \times (x - 2)^2 \times (x + 10)$
2. $4x^2 + 12x + 9$ and $4x^2 - 9$
3. $2x^2 - 18$ and $5x^3 + 30x^2 + 45x$

find the sum.
4. $\frac{6y - 4}{y^2 - 5} + \frac{3y + 1}{y^2 - 5}$
5. $\frac{x + 2}{x^2 + 4x + 4} + \frac{2}{x + 2}$
6. $\frac{4}{y^2 - 25} + \frac{6}{x^2 + 8x + 5}$

Question

adding and subtracting rational expressions
find the lcm for each group of expressions.
1. $2x^2 - 8x + 8$ and $3x^2 + 27x - 30$
$2x^2 - 8x + 8 = 2(x^2 - 4x + 4) = 2(x - 2)^2$
$3x^2 + 27x - 30 = 3(x^2 + 9x - 10) = 3(x + 10)(x - 1)$
$2 \times 3 \times (x - 2)^2 \times (x + 10) \times (x - 1)$
the lcm = $2 \times 3 \times (x - 2)^2 \times (x + 10)$
2. $4x^2 + 12x + 9$ and $4x^2 - 9$
3. $2x^2 - 18$ and $5x^3 + 30x^2 + 45x$

find the sum.
4. $\frac{6y - 4}{y^2 - 5} + \frac{3y + 1}{y^2 - 5}$
5. $\frac{x + 2}{x^2 + 4x + 4} + \frac{2}{x + 2}$
6. $\frac{4}{y^2 - 25} + \frac{6}{x^2 + 8x + 5}$

Accepted Answer

### Problem 2: Find the LCM of $4x^2 + 12x + 9$ and $4x^2 - 9$
# Explanation:
## Step 1: Factor the first expression
Factor $4x^2 + 12x + 9$ using the perfect square formula $(a + b)^2 = a^2 + 2ab + b^2$, where $a = 2x$ and $b = 3$.
$4x^2 + 12x + 9=(2x + 3)^2$
## Step 2: Factor the second expression
Factor $4x^2 - 9$ using the difference of squares formula $a^2 - b^2=(a + b)(a - b)$, where $a = 2x$ and $b = 3$.
$4x^2 - 9=(2x + 3)(2x - 3)$
## Step 3: Find the LCM
The LCM of two polynomials is the product of the highest powers of all factors that appear in the factorizations.
The factors are $(2x + 3)^2$ (from the first polynomial) and $(2x - 3)$ (from the second polynomial).
So, $LCM=(2x + 3)^2(2x - 3)$
# Answer:
$(2x + 3)^2(2x - 3)$

### Problem 3: Find the LCM of $2x^2 - 18$ and $5x^3 + 30x^2 + 45x$
# Explanation:
## Step 1: Factor the first expression
First, factor out the greatest common factor (GCF) of $2x^2 - 18$, which is 2. Then use the difference of squares formula $a^2 - b^2=(a + b)(a - b)$ where $a = x$ and $b = 3$.
$2x^2 - 18 = 2(x^2 - 9)=2(x + 3)(x - 3)$
## Step 2: Factor the second expression
Factor out the GCF of $5x^3 + 30x^2 + 45x$, which is $5x$. Then factor the quadratic expression inside the parentheses using the perfect square formula $(a + b)^2=a^2 + 2ab + b^2$ where $a = x$ and $b = 3$.
$5x^3 + 30x^2 + 45x=5x(x^2 + 6x + 9)=5x(x + 3)^2$
## Step 3: Find the LCM
The LCM is the product of the highest powers of all factors: $2$, $5x$, $(x + 3)^2$, and $(x - 3)$.
So, $LCM = 2\times5x(x + 3)^2(x - 3)=10x(x + 3)^2(x - 3)$
# Answer:
$10x(x + 3)^2(x - 3)$

### Problem 4: Find the sum $\frac{6y - 4}{y^2 - 5}+\frac{3y + 1}{y^2 - 5}$
# Explanation:
## Step 1: Add the numerators (same denominator)
Since the denominators are the same ($y^2 - 5$), we add the numerators: $(6y - 4)+(3y + 1)$
## Step 2: Simplify the numerator
Combine like terms: $6y+3y-4 + 1=9y-3$
## Step 3: Write the result
The sum is $\frac{9y - 3}{y^2 - 5}$. We can also factor out a 3 from the numerator: $\frac{3(3y - 1)}{y^2 - 5}$
# Answer:
$\frac{9y - 3}{y^2 - 5}$ (or $\frac{3(3y - 1)}{y^2 - 5}$)

### Problem 5: Find the sum $\frac{x + 2}{x^2 + 4x + 4}+\frac{2}{x + 2}$
# Explanation:
## Step 1: Factor the denominator of the first fraction
Factor $x^2 + 4x + 4$ using the perfect square formula $(a + b)^2=a^2 + 2ab + b^2$ where $a = x$ and $b = 2$.
$x^2 + 4x + 4=(x + 2)^2$
So the first fraction becomes $\frac{x + 2}{(x + 2)^2}=\frac{1}{x + 2}$ (after canceling one $(x + 2)$ term, $x
eq - 2$)
## Step 2: Add the two fractions
Now we have $\frac{1}{x + 2}+\frac{2}{x + 2}$. Since the denominators are the same, we add the numerators: $1+2 = 3$
## Step 3: Write the result
The sum is $\frac{3}{x + 2}$
# Answer:
$\frac{3}{x + 2}$

### Problem 6: Find the sum $\frac{4}{y^2 - 25}+\frac{6}{x^2 + 6x + 5}$ (Note: There seems to be a typo, assuming it's $\frac{4}{x^2 - 25}+\frac{6}{x^2 + 6x + 5}$ for consistency)
# Explanation:
## Step 1: Factor the denominators
- Factor $x^2 - 25$ using the difference of squares formula: $x^2 - 25=(x + 5)(x - 5)$
- Factor $x^2 + 6x + 5$ using the formula $x^2+(a + b)x+ab=(x + a)(x + b)$ where $a = 1$ and $b = 5$. So $x^2 + 6x + 5=(x + 1)(x + 5)$
## Step 2: Find the LCM of the denominators
The LCM of $(x + 5)(x - 5)$ and $(x + 1)(x + 5)$ is $(x + 5)(x - 5)(x + 1)$
## Step 3: Rewrite the fractions with the LCM as the denominator
- For $\frac{4}{(x + 5)(x - 5)}$, multiply numerator and denominator by $(x + 1)$: $\frac{4(x + 1)}{(x + 5)(x - 5)(x + 1)}$
- For $\frac{6}{(x + 1)(x + 5)}$, multiply numerator and denominator by $(x - 5)$: $\frac{6(x - 5)}{(x + 5)(x - 5)(x + 1)}$
## Step 4: Add the numerators
$4(x + 1)+6(x - 5)=4x+4 + 6x-30=10x - 26$
## Step 5: Write the result
The sum is $\frac{10x - 26}{(x + 5)(x - 5)(x + 1)}$. We can factor out a 2 from the numerator: $\frac{2(5x - 13)}{(x + 5)(x - 5)(x + 1)}$
# Answer:
$\frac{10x - 26}{(x + 5)(x - 5)(x + 1)}$ (or $\frac{2(5x - 13)}{(x + 5)(x - 5)(x + 1)}$)

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QUESTION IMAGE

Question

Explanation:

Problem 2: Find the LCM of \(4x^2 + 12x + 9\) and \(4x^2 - 9\)

Step 1: Factor the first expression

Step 2: Factor the second expression

Step 3: Find the LCM

Step 1: Factor the first expression

Step 2: Factor the second expression

Step 3: Find the LCM

Step 1: Add the numerators (same denominator)

Step 2: Simplify the numerator

Step 3: Write the result

Answer:

Problem 3: Find the LCM of \(2x^2 - 18\) and \(5x^3 + 30x^2 + 45x\)