Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{\frac{9 x^{2}+3 x-2}{12 x^{2}+5 x-2}}{\frac{9 x^{2}-6 x+1}{8 x^{2}-10 x-3}} $$

Short answer

\( \frac{(4x - 3)(2x + 1)}{(4x - 1)(3x - 1)} \)

Step-by-step solution

Understand the given expression

The given expression is \ \[ \frac{\frac{9 x^{2}+3 x-2}{12 x^{2}+5 x-2}}{\frac{9 x^{2}-6 x+1}{8 x^{2}-10 x -3}} \] \ This can be seen as a division of two fractions. To simplify, we will first convert the division into multiplication by taking the reciprocal of the second fraction.

Convert division to multiplication

Convert the division into multiplication by taking the reciprocal of the second fraction. This gives us: \ \[ \frac{9 x^{2}+3 x-2}{12 x^{2}+5 x-2} \times \frac{8 x^{2}-10 x-3}{9 x^{2}-6 x+1} \] \ Now we can proceed with multiplying the numerators together and the denominators together.

Factorize the numerators and denominators

Factorize the numerators and denominators where possible: \ For the first fraction: \ \( 9x^2 + 3x - 2 \ = \ (3x + 2)(3x - 1) \) \ \( 12x^2 + 5x - 2 \ = \ (3x + 2)(4x - 1) \) \ For the second fraction: \ \( 9x^2 - 6x + 1 \ = \ (3x - 1)^2 \) \ \( 8x^2 - 10x - 3 \ = \ (4x - 3)(2x + 1) \)

Substitute the factored forms

Substitute the factored forms back into the expression: \ \[ \frac{(3x + 2)(3x - 1)}{(3x + 2)(4x - 1)} \times \frac{(4x - 3)(2x + 1)}{(3x - 1)^2} \]

Simplify the expression

Cancel out common factors from the numerators and denominators: \ \( 3x + 2 \) cancels out, and one \( 3x - 1 \) from the numerator and denominator also cancels out: \ \[ \frac{(4x - 3)(2x + 1)}{(4x - 1)(3x - 1)} \] \ This is the simplified and factored form of the expression.

Key concepts

Factoring Polynomials

Before diving into simplifying rational expressions, it's essential to be familiar with factoring polynomials. Polynomials are algebraic expressions that include variables raised to various powers. Factoring them simplifies these expressions by breaking them into products of simpler expressions.

To factor a polynomial like \( 9x^2 + 3x - 2 \), we look for two binomials that, when multiplied, reproduce the original polynomial. For instance, \( 9x^2 + 3x - 2 \) factors into \( (3x + 2)(3x - 1) \). You can verify this by distributing the binomials: \( (3x + 2)(3x - 1) = 9x^2 - 3x + 6x - 2 = 9x^2 + 3x - 2 \). Each term results from combining like terms.

When you factor polynomials in a rational expression, you simplify the overall expression, making it easier to find and cancel common factors.

Reciprocal Operation

The process of simplifying rational expressions often involves reciprocal operations, especially when division is present. The reciprocal of a given fraction is simply flipping its numerator and denominator.

For example, given the division of two rational expressions: \( \frac{9 x^{2}+3 x-2}{12 x^{2}+5 x-2} \big/ \frac{9 x^{2}-6 x+1}{8 x^{2}-10 x-3} \), you can replace the division with multiplication by taking the reciprocal of the second fraction. This expression becomes: \( \frac{9 x^{2}+3 x-2}{12 x^{2}+5 x-2} \times \frac{8 x^{2}-10 x-3}{9 x^{2}-6 x+1} \).

Using the reciprocal makes it straightforward to multiply rational expressions instead of performing division, aiding in simplifying the overall equation.

Multiplying Rational Expressions

Once the division is converted to multiplication using the reciprocal, the next step is multiplying the rational expressions. This involves multiplying the numerators together and the denominators together.

For the expression: \( \frac{9 x^{2}+3 x-2}{12 x^{2}+5 x-2} \times \frac{8 x^{2}-10 x-3}{9 x^{2}-6 x+1} \), you factor each numerator and denominator:

• \( 9x^2 + 3x - 2 = (3x + 2)(3x - 1) \)
• \( 12x^2 + 5x - 2 = (3x + 2)(4x - 1) \)
• \( 9x^2 - 6x + 1 = (3x - 1)^2 \)
• \( 8x^2 - 10x - 3 = (4x - 3)(2x + 1) \)

Then the expression becomes: \( \frac{(3x + 2)(3x - 1)}{(3x + 2)(4x - 1)} \times \frac{(4x - 3)(2x + 1)}{(3x - 1)^2} \).

Multiplying rational expressions this way sets the stage for canceling common factors.

Canceling Common Factors

The last step in simplifying rational expressions is to cancel common factors, which appear in both the numerator and the denominator. This reduces the expression to its simplest form.

For instance, in the expression after factoring and multiplying: \( \frac{(3x + 2)(3x - 1)}{(3x + 2)(4x - 1)} \times \frac{(4x - 3)(2x + 1)}{(3x - 1)^2} \),
you can cancel out:\

• The common factor \( 3x + 2 \) in the numerator and denominator
• One instance of \( 3x - 1 \) in the numerator and denominator

This simplifies to: \( \frac{(4x - 3)(2x + 1)}{(4x - 1)(3x - 1)} \).

Canceling common factors is crucial because it reduces the expression, making it simpler to understand and solve.