Question

Simplify the expression. $$\frac{x+8}{3 x-1}+\frac{x+3}{x+1}$$

Short answer

\(\frac{4x^2 + 8x + 5}{(3x - 1)(x + 1)}\)

Step-by-step solution

Find the Common Denominator

To combine these fractions, we first need to find the common denominator. The common denominator of the fractions is the least common multiple (LCM) of the two denominators. In this case, the LCM of \(3x - 1\) and \(x + 1\) is the product of these two polynomials: \( (3x - 1) \cdot (x + 1) \).

Make Equivalent Fractions

The next step is to rewrite both fractions as equivalent fractions with the common denominator. For the first fraction, multiply the numerator and denominator by \((x + 1)\) so the denominator is \((3x - 1) \cdot (x + 1)\). For the second fraction, multiply the numerator and denominator by \((3x - 1)\) so it will also have denominator \((3x - 1) \cdot (x + 1)\). This yields \(\frac{x+8}{3 x-1} \cdot \frac{x+1}{x+1}+\frac{x+3}{x+1} \cdot \frac{3x-1}{3x-1} = \frac{(x+8)(x+1)}{(3x-1)(x+1)}+\frac{(x+3)(3x-1)}{(3x-1)(x+1)}\).

Combine the Fractions

Now that the two fractions have the same denominator, they can be combined. Add the two fractions together: \(\frac{(x+8)(x+1)+(x+3)(3x-1)}{(3x-1)(x+1)}\).

Simplify the Numerator

Expand the expressions in the numerator to further simplify the expression: \(\frac{x^2 + 9x + 8 + 3x^2 - x - 3}{(3x - 1)(x + 1)} = \frac{4x^2 + 8x + 5}{(3x - 1)(x + 1)}\).

Final Simplified Expression

After simplifying, the final result of the expression is \(\frac{4x^2 + 8x + 5}{(3x - 1)(x + 1)}\). This fraction cannot be simplified any further, as the numerator does not factorize with the denominator.