Questions: Simplify. (u^2+3u+2)/(3u^2-12)

Transcript text: Simplify. \[ \frac{u^{2}+3 u+2}{3 u^{2}-12} \]

Solution

Solution Steps

Step 1: Factor the numerator and denominator

First, factor the numerator and the denominator of the given expression.

The numerator is: \[ u^{2} + 3u + 2 \] This can be factored as: \[ (u + 1)(u + 2) \]

The denominator is: \[ 3u^{2} - 12 \] Factor out the greatest common factor (GCF), which is 3: \[ 3(u^{2} - 4) \] Now, factor \(u^{2} - 4\) as a difference of squares: \[ 3(u - 2)(u + 2) \]

Step 2: Rewrite the expression with factored forms

Substitute the factored forms of the numerator and denominator back into the original expression: \[ \frac{(u + 1)(u + 2)}{3(u - 2)(u + 2)} \]

Step 3: Cancel common factors

Notice that \((u + 2)\) appears in both the numerator and the denominator. Cancel this common factor: \[ \frac{u + 1}{3(u - 2)} \]

Final Answer

The simplified form of the expression is: \[ \boxed{\frac{u + 1}{3(u - 2)}} \]

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