Find the Domain (5x+38x+21)/(3x^2+22x+7)

Problem

(5*x+38*x+21)/(3*x2+22*x+7)

Solution

1. Identify the condition for the domain of a rational function. The domain consists of all real numbers except those that make the denominator equal to zero.

2. Set the denominator equal to zero to find the excluded values.

3*x2+22*x+7=0

3. Factor the quadratic expression in the denominator. We look for two numbers that multiply to 3×7=21 and add to 22 which are 21 and 1

3*x2+21*x+x+7=0

3*x*(x+7)+1*(x+7)=0

(3*x+1)*(x+7)=0

4. Solve for x by setting each factor to zero.

3*x+1=0⇒x=−1/3

x+7=0⇒x=−7

5. State the domain by excluding these values from the set of all real numbers.

Final Answer

Domain={x|x∈ℝ,x≠−7,x≠−1/3}

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