Simplify (4x^2-4x-3)/(8x^3+1)

Solution:

Step 1: Factor the numerator by grouping.

• Step 1.1: For a quadratic expression $ax^2 + bx + c$, split the middle term into two terms whose product equals $ac = 4 \cdot (-3) = -12$ and whose sum equals $b = -4$.

  • Step 1.1.1: Extract $-4$ from $-4x$. $\frac{4x^2 - 4(x) - 3}{8x^3 + 1}$
  • Step 1.1.2: Decompose $-4$ into $2$ and $-6$. $\frac{4x^2 + (2 - 6)x - 3}{8x^3 + 1}$
  • Step 1.1.3: Use the distributive property. $\frac{4x^2 + 2x - 6x - 3}{8x^3 + 1}$

• Step 1.2: Factor out the greatest common factor (GCF) from each pair of terms.

  • Step 1.2.1: Pair the terms for grouping. $\frac{(4x^2 + 2x) - (6x + 3)}{8x^3 + 1}$
  • Step 1.2.2: Factor the GCF from each group. $\frac{2x(2x + 1) - 3(2x + 1)}{8x^3 + 1}$

• Step 1.3: Factor by taking out the common factor $2x + 1$. $\frac{(2x + 1)(2x - 3)}{8x^3 + 1}$

Step 2: Simplify the denominator.

• Step 2.1: Express $8x^3$ as $(2x)^3$. $\frac{(2x + 1)(2x - 3)}{(2x)^3 + 1}$

• Step 2.2: Represent $1$ as $1^3$. $\frac{(2x + 1)(2x - 3)}{(2x)^3 + 1^3}$

• Step 2.3: Factor using the sum of cubes formula $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ where $a = 2x$ and $b = 1$. $\frac{(2x + 1)(2x - 3)}{(2x + 1)(4x^2 - 2x + 1)}$

• Step 2.4: Simplify the expression.

  • Step 2.4.1: Apply the square to $2x$. $\frac{(2x + 1)(2x - 3)}{(2x + 1)(4x^2 - 2x + 1)}$
  • Step 2.4.2: Calculate $2^2$. $\frac{(2x + 1)(2x - 3)}{(2x + 1)(4x^2 - 2x + 1)}$
  • Step 2.4.3: Multiply $2$ by $-1$. $\frac{(2x + 1)(2x - 3)}{(2x + 1)(4x^2 - 2x + 1)}$
  • Step 2.4.4: Multiply $-2$ by $1$. $\frac{(2x + 1)(2x - 3)}{(2x + 1)(4x^2 - 2x + 1)}$
  • Step 2.4.5: Recognize that any number raised to the power of zero is one. $\frac{(2x + 1)(2x - 3)}{(2x + 1)(4x^2 - 2x + 1)}$

Step 3: Cancel the common factor.

• Step 3.1: Eliminate the common factor. $\frac{\cancel{(2x + 1)}(2x - 3)}{\cancel{(2x + 1)}(4x^2 - 2x + 1)}$
• Step 3.2: Write the simplified expression. $\frac{2x - 3}{4x^2 - 2x + 1}$

Knowledge Notes:

To simplify a rational expression, you can follow these steps:

1. Factor by Grouping: This technique is used when you have a polynomial with four terms. You group the terms into pairs and factor out the greatest common factor from each pair. If done correctly, you should be able to factor out a common binomial factor from the grouped terms.

2. Factor the Numerator and Denominator: Factoring the numerator and denominator separately can help in simplifying the expression by canceling out common factors.

3. Sum of Cubes: The sum of cubes is a special factoring formula that applies to two terms that are both perfect cubes. The formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

4. Cancel Common Factors: Once you have factored the numerator and the denominator, cancel out any common factors that appear in both. This is only possible if the factors are multiplied, not if they are added or subtracted.

5. Simplify the Expression: After canceling out common factors, rewrite the expression in its simplest form.

Remember that you cannot cancel terms across addition or subtraction; you can only cancel factors that are common to the numerator and the denominator.