Questions: Simplify the rational expression. (2x^2 - 3x - 2) / (6x^3 + 3x^2 + 2x + 1)

Transcript text: Simplify the rational expression. \[ \frac{2 x^{2}-3 x-2}{6 x^{3}+3 x^{2}+2 x+1} \]

Solution

Solution Steps

To simplify the given rational expression, we need to factor both the numerator and the denominator and then cancel out any common factors. The numerator is a quadratic expression, and the denominator is a cubic expression. We will use techniques such as factoring by grouping or using the quadratic formula for the numerator and synthetic division or polynomial division for the denominator.

Step 1: Factor the Numerator

The numerator \( 2x^2 - 3x - 2 \) can be factored as: \[ 2x^2 - 3x - 2 = (x - 2)(2x + 1) \]

Step 2: Factor the Denominator

The denominator \( 6x^3 + 3x^2 + 2x + 1 \) can be factored as: \[ 6x^3 + 3x^2 + 2x + 1 = (2x + 1)(3x^2 + 1) \]

Step 3: Simplify the Rational Expression

Now, we can write the rational expression as: \[ \frac{2x^2 - 3x - 2}{6x^3 + 3x^2 + 2x + 1} = \frac{(x - 2)(2x + 1)}{(2x + 1)(3x^2 + 1)} \] We can cancel the common factor \( (2x + 1) \): \[ \frac{(x - 2)}{(3x^2 + 1)} \]

Final Answer

The simplified expression is: \[ \boxed{\frac{x - 2}{3x^2 + 1}} \]

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