Simplify (4s^2+20s+24)/(16s^2-32s-128)

Solution:

Step 1: Extract the common factor from the numerator and denominator.

Step 1.1: Extract the factor of 4 from the term $4s^2$.

$\frac{4(s^2) + 20s + 24}{16s^2 - 32s - 128}$

Step 1.2: Extract the factor of 4 from the term $20s$.

$\frac{4(s^2) + 4(5s) + 24}{16s^2 - 32s - 128}$

Step 1.3: Extract the factor of 4 from the sum $4(s^2) + 4(5s)$.

$\frac{4(s^2 + 5s) + 24}{16s^2 - 32s - 128}$

Step 1.4: Extract the factor of 4 from the constant 24.

$\frac{4(s^2 + 5s) + 4 \cdot 6}{16s^2 - 32s - 128}$

Step 1.5: Extract the factor of 4 from the entire numerator.

$\frac{4(s^2 + 5s + 6)}{16s^2 - 32s - 128}$

Step 1.6: Simplify by canceling out the common factors.

Step 1.6.1: Extract the factor of 4 from $16s^2$.

$\frac{4(s^2 + 5s + 6)}{4(4s^2) - 32s - 128}$

Step 1.6.2: Extract the factor of 4 from $-32s$.

$\frac{4(s^2 + 5s + 6)}{4(4s^2) + 4(-8s) - 128}$

Step 1.6.3: Extract the factor of 4 from the sum $4(4s^2) + 4(-8s)$.

$\frac{4(s^2 + 5s + 6)}{4(4s^2 - 8s) - 128}$

Step 1.6.4: Extract the factor of 4 from $-128$.

$\frac{4(s^2 + 5s + 6)}{4(4s^2 - 8s) + 4(-32)}$

Step 1.6.5: Extract the factor of 4 from the entire denominator.

$\frac{4(s^2 + 5s + 6)}{4(4s^2 - 8s - 32)}$

Step 1.6.6: Cancel the common factor of 4.

$\frac{\cancel{4}(s^2 + 5s + 6)}{\cancel{4}(4s^2 - 8s - 32)}$

Step 1.6.7: Rewrite the simplified expression.

$\frac{s^2 + 5s + 6}{4s^2 - 8s - 32}$

Step 2: Factor the numerator using the AC method.

Step 2.1: Identify integers with a product of 6 and a sum of 5.

$2, 3$

Step 2.2: Write the factored form of the numerator.

$\frac{(s + 2)(s + 3)}{4s^2 - 8s - 32}$

Step 3: Simplify the denominator.

Step 3.1: Extract the factor of 4 from the denominator.

Step 3.1.1: Extract the factor of 4 from $4s^2$.

$\frac{(s + 2)(s + 3)}{4(s^2) - 8s - 32}$

Step 3.1.2: Extract the factor of 4 from $-8s$.

$\frac{(s + 2)(s + 3)}{4(s^2) + 4(-2s) - 32}$

Step 3.1.3: Extract the factor of 4 from $-32$.

$\frac{(s + 2)(s + 3)}{4(s^2 - 2s - 8)}$

Step 3.1.4: Extract the factor of 4 from the sum $4(s^2 - 2s)$.

$\frac{(s + 2)(s + 3)}{4(s^2 - 2s - 8)}$

Step 3.2: Factor the quadratic expression within the denominator.

Step 3.2.1: Identify integers with a product of -8 and a sum of -2.

$-4, 2$

Step 3.2.2: Write the factored form of the denominator.

$\frac{(s + 2)(s + 3)}{4((s - 4)(s + 2))}$

Step 4: Cancel the common factor of $s + 2$.

Step 4.1: Cancel the common factor.

$\frac{\cancel{(s + 2)}(s + 3)}{4(s - 4)\cancel{(s + 2)}}$

Step 4.2: Rewrite the final simplified expression.

$\frac{s + 3}{4(s - 4)}$

Knowledge Notes:

To simplify a rational expression, we follow these steps:

1. Factorization: Break down both the numerator and denominator into their factors. This can involve taking out common factors or using methods such as the AC method to factor quadratic expressions.

2. Common Factors: Look for and cancel out any common factors that appear in both the numerator and the denominator.

3. AC Method: This is a technique used to factor quadratic expressions of the form $ax^2 + bx + c$. The method involves finding two numbers that multiply to $a \cdot c$ and add up to $b$. These two numbers are then used to split the middle term and factor by grouping.

4. Simplification: After canceling common factors, rewrite the expression to its simplest form.

5. Checking: Always check that the factors you have canceled do not include any restrictions on the variable (values that would make the denominator zero are not allowed).

By following these steps, we can simplify complex rational expressions into their simplest forms.