Simplify (12x^2+2x-2)/(27x^3-1)
The problem is asking to perform algebraic simplification on the given rational expression. The expression consists of a polynomial in the numerator, which is a quadratic polynomial (12x^2 + 2x - 2), and a polynomial in the denominator, which is a cubic polynomial (27x^3 - 1). The task is to simplify this fraction by factoring both the numerator and the denominator if possible and then reducing it by canceling out any common factors that may exist between them. The goal is to rewrite the complex expression in a simpler or more easily understandable form without changing its value.
$\frac{12 x^{2} + 2 x - 2}{27 x^{3} - 1}$
Answer
Solution:
Step 1: Refactor the Numerator
Step 1.1: Extract the Common Factor
Extract the factor of $2$ from the terms in the numerator.
$$\frac{2(6x^2) + 2x - 2}{27x^3 - 1}$$
Step 1.1.1: Factor Out from $12x^2$
$$\frac{2(6x^2) + 2x - 2}{27x^3 - 1}$$
Step 1.1.2: Factor Out from $2x$
$$\frac{2(6x^2) + 2(x) - 2}{27x^3 - 1}$$
Step 1.1.3: Factor Out from $-2$
$$\frac{2(6x^2) + 2(x) + 2(-1)}{27x^3 - 1}$$
Step 1.1.4: Combine the Factored Terms
$$\frac{2(6x^2 + x - 1)}{27x^3 - 1}$$
Step 1.2: Grouping for Factoring
Step 1.2.1: Split the Middle Term
Split $b$ into two numbers whose product is $ac$ ($6 \cdot -1 = -6$) and whose sum is $b$ ($1$).
$$\frac{2(6x^2 + 1x - 1)}{27x^3 - 1}$$
Step 1.2.1.1: Use Identity
$$\frac{2(6x^2 + 1x - 1)}{27x^3 - 1}$$
Step 1.2.1.2: Decompose $1x$
$$\frac{2(6x^2 + (-2 + 3)x - 1)}{27x^3 - 1}$$
Step 1.2.1.3: Distribute $x$
$$\frac{2(6x^2 - 2x + 3x - 1)}{27x^3 - 1}$$
Step 1.2.2: Factor by Grouping
Step 1.2.2.1: Create Groups
$$\frac{2((6x^2 - 2x) + (3x - 1))}{27x^3 - 1}$$
Step 1.2.2.2: Factor Out GCF from Each Group
$$\frac{2(2x(3x - 1) + 1(3x - 1))}{27x^3 - 1}$$
Step 1.2.3: Factor Out the Common Binomial
$$\frac{2(3x - 1)(2x + 1)}{27x^3 - 1}$$
Step 2: Simplify the Denominator
Step 2.1: Express as a Cube
$$\frac{2(3x - 1)(2x + 1)}{(3x)^3 - 1}$$
Step 2.2: Express $1$ as a Cube
$$\frac{2(3x - 1)(2x + 1)}{(3x)^3 - 1^3}$$
Step 2.3: Factor Using Difference of Cubes
$$\frac{2(3x - 1)(2x + 1)}{(3x - 1)(9x^2 + 3x + 1)}$$
Step 2.4: Simplify the Expression
Step 2.4.1: Apply the Power Rule
$$\frac{2(3x - 1)(2x + 1)}{(3x - 1)(3^2x^2 + 3x \cdot 1 + 1^2)}$$
Step 2.4.2: Compute the Square of $3$
$$\frac{2(3x - 1)(2x + 1)}{(3x - 1)(9x^2 + 3x \cdot 1 + 1^2)}$$
Step 2.4.3: Multiply $3$ and $1$
$$\frac{2(3x - 1)(2x + 1)}{(3x - 1)(9x^2 + 3x + 1)}$$
Step 2.4.4: Simplify the Constant Term
$$\frac{2(3x - 1)(2x + 1)}{(3x - 1)(9x^2 + 3x + 1)}$$
Step 3: Cancel the Common Factor
Step 3.1: Eliminate the Common Binomial
$$\frac{2 \cancel{(3x - 1)}(2x + 1)}{\cancel{(3x - 1)}(9x^2 + 3x + 1)}$$
Step 3.2: Final Simplified Expression
$$\frac{2(2x + 1)}{9x^2 + 3x + 1}$$
Knowledge Notes:
The problem involves simplifying a rational expression by factoring the numerator and the denominator and then canceling out common factors. The key knowledge points include:
Factoring out common factors: This is a technique used to simplify expressions by finding and extracting common factors from terms.
Factoring by grouping: A method used to factor polynomials that involves rearranging the terms and factoring out the greatest common factor from each pair of terms.
Difference of cubes: A special factoring formula used when a polynomial is in the form of $a^3 - b^3$, which can be factored into $(a - b)(a^2 + ab + b^2)$.
Canceling common factors: In a rational expression, if the same factor appears in both the numerator and the denominator, it can be canceled out to simplify the expression.
The process of simplifying the given rational expression involves these steps, ultimately leading to the cancellation of the common binomial factor $(3x - 1)$ and yielding the simplified form of the expression.
