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

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Solution:

Step 1: Refactor the Numerator

Step 1.1: Extract the Common Factor

Extract the factor of $2$ from the terms in the numerator.

$\frac{2 (6 x^{2}) + 2 x - 2}{27 x^{3} - 1}$

Step 1.1.1: Factor Out from $12 x^{2}$

$\frac{2 (6 x^{2}) + 2 x - 2}{27 x^{3} - 1}$

Step 1.1.2: Factor Out from $2 x$

$\frac{2 (6 x^{2}) + 2 (x) - 2}{27 x^{3} - 1}$

Step 1.1.3: Factor Out from $- 2$

$\frac{2 (6 x^{2}) + 2 (x) + 2 (- 1)}{27 x^{3} - 1}$

Step 1.1.4: Combine the Factored Terms

$\frac{2 (6 x^{2} + x - 1)}{27 x^{3} - 1}$

Step 1.2: Grouping for Factoring

Step 1.2.1: Split the Middle Term

Split $b$ into two numbers whose product is $a c$ ( $6 \cdot - 1 = - 6$ ) and whose sum is $b$ ( $1$ ).

$\frac{2 (6 x^{2} + 1 x - 1)}{27 x^{3} - 1}$

Step 1.2.1.1: Use Identity

$\frac{2 (6 x^{2} + 1 x - 1)}{27 x^{3} - 1}$

Step 1.2.1.2: Decompose $1 x$

$\frac{2 (6 x^{2} + (- 2 + 3) x - 1)}{27 x^{3} - 1}$

Step 1.2.1.3: Distribute $x$

$\frac{2 (6 x^{2} - 2 x + 3 x - 1)}{27 x^{3} - 1}$

Step 1.2.2: Factor by Grouping

Step 1.2.2.1: Create Groups

$\frac{2 ((6 x^{2} - 2 x) + (3 x - 1))}{27 x^{3} - 1}$

Step 1.2.2.2: Factor Out GCF from Each Group

$\frac{2 (2 x (3 x - 1) + 1 (3 x - 1))}{27 x^{3} - 1}$

Step 1.2.3: Factor Out the Common Binomial

$\frac{2 (3 x - 1) (2 x + 1)}{27 x^{3} - 1}$

Step 2: Simplify the Denominator

Step 2.1: Express as a Cube

$\frac{2 (3 x - 1) (2 x + 1)}{(3 x)^{3} - 1}$

Step 2.2: Express $1$ as a Cube

$\frac{2 (3 x - 1) (2 x + 1)}{(3 x)^{3} - 1^{3}}$

Step 2.3: Factor Using Difference of Cubes

$\frac{2 (3 x - 1) (2 x + 1)}{(3 x - 1) (9 x^{2} + 3 x + 1)}$

Step 2.4: Simplify the Expression

Step 2.4.1: Apply the Power Rule

$\frac{2 (3 x - 1) (2 x + 1)}{(3 x - 1) (3^{2} x^{2} + 3 x \cdot 1 + 1^{2})}$

Step 2.4.2: Compute the Square of $3$

$\frac{2 (3 x - 1) (2 x + 1)}{(3 x - 1) (9 x^{2} + 3 x \cdot 1 + 1^{2})}$

Step 2.4.3: Multiply $3$ and $1$

$\frac{2 (3 x - 1) (2 x + 1)}{(3 x - 1) (9 x^{2} + 3 x + 1)}$

Step 2.4.4: Simplify the Constant Term

$\frac{2 (3 x - 1) (2 x + 1)}{(3 x - 1) (9 x^{2} + 3 x + 1)}$

Step 3: Cancel the Common Factor

Step 3.1: Eliminate the Common Binomial

$\frac{2 (3 x - 1) (2 x + 1)}{(3 x - 1) (9 x^{2} + 3 x + 1)}$

Step 3.2: Final Simplified Expression

$\frac{2 (2 x + 1)}{9 x^{2} + 3 x + 1}$

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 $(a - b) (a^{2} + a b + b^{2})$ .
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 $(3 x - 1)$ and yielding the simplified form of the expression.

Simplify (12x^2+2x-2)/(27x^3-1)

Answer

Solution:

Step 1: Refactor the Numerator

Step 1.1: Extract the Common Factor

Step 1.1.1: Factor Out from 12x2

Step 1.1.2: Factor Out from 2x

Step 1.1.3: Factor Out from −2

Step 1.1.4: Combine the Factored Terms

Step 1.2: Grouping for Factoring

Step 1.2.1: Split the Middle Term

Step 1.2.1.1: Use Identity

Step 1.2.1.2: Decompose 1x

Step 1.2.1.3: Distribute x

Step 1.2.2: Factor by Grouping

Step 1.2.2.1: Create Groups

Step 1.2.2.2: Factor Out GCF from Each Group

Step 1.2.3: Factor Out the Common Binomial

Step 2: Simplify the Denominator

Step 2.1: Express as a Cube

Step 2.2: Express 1 as a Cube

Step 2.3: Factor Using Difference of Cubes

Step 2.4: Simplify the Expression

Step 2.4.1: Apply the Power Rule

Step 2.4.2: Compute the Square of 3

Step 2.4.3: Multiply 3 and 1

Step 2.4.4: Simplify the Constant Term

Step 3: Cancel the Common Factor

Step 3.1: Eliminate the Common Binomial

Step 3.2: Final Simplified Expression

Knowledge Notes:

Popular Problems

Step 1.1.1: Factor Out from $12 x^{2}$

Step 1.1.2: Factor Out from $2 x$

Step 1.1.3: Factor Out from $- 2$

Step 1.2.1.2: Decompose $1 x$

Step 1.2.1.3: Distribute $x$

Step 2.2: Express $1$ as a Cube

Step 2.4.2: Compute the Square of $3$

Step 2.4.3: Multiply $3$ and $1$