To rationalize the denominator of \( h(x) = \frac{3x + 5}{\sqrt{2x + 1}} \), we multiply both the numerator and the denominator by \( \sqrt{2x + 1} \) to eliminate the square root in the denominator:

["Rationalizing the Denominator in Rational Functions: How to Simplify ( h(x) = \frac{3x + 5}{\sqrt{2x + 1}} )", "When working with rational functions that contain square roots in the denominator, one common refinement is rationalizing the denominator. This process eliminates irrational expressions from the denominator, making the expression cleaner, easier to work with, and preferred in algebra and calculus. In this article, we explain why and how to rationalize the denominator of the function:", "[\nh(x) = \frac{3x + 5}{\sqrt{2x + 1}}\n]", "### Why Rationalize the Denominator?", "A denominator with a square root introduces complexity in further calculations, such as differentiation, integration, or comparison with other expressions. Rationalizing simplifies expressions and ensures standard mathematical form, especially useful when evaluating limits or performing algebraic manipulations.", "---", "### The Method: Multiply Numerator and Denominator by ( \sqrt{2x + 1} )", "To rationalize the denominator of ( h(x) = \frac{3x + 5}{\sqrt{2x + 1}} ), we eliminate the square root in the denominator by multiplying both the numerator and the denominator by ( \sqrt{2x + 1} ). This technique works because:", "[\n\sqrt{2x + 1} \ imes \sqrt{2x + 1} = 2x + 1\n]", "Now, let’s apply this step-by-step:", "[\nh(x) = \frac{3x + 5}{\sqrt{2x + 1}} \cdot \frac{\sqrt{2x + 1}}{\sqrt{2x + 1}}\n]", "Apply the multiplication:", "[\n= \frac{(3x + 5) \cdot \sqrt{2x + 1}}{2x + 1}\n]", "The denominator is now a rational expression with no square root: ( 2x + 1 ). The numerator remains written with the square root to reflect the original function's form.", "---", "### Result After Rationalization", "[\nh(x) = \frac{(3x + 5)\sqrt{2x + 1}}{2x + 1}\n]", "This is the fully rationalized form of the function. The denominator is now a clean polynomial, facilitating further algebraic operations and analysis.", "---", "### Summary", "- Original function: ( h(x) = \frac{3x + 5}{\sqrt{2x + 1}} )\n- Action: Multiply numerator and denominator by ( \sqrt{2x + 1} )\n- Result: ( h(x) = \frac{(3x + 5)\sqrt{2x + 1}}{2x + 1} )", "Rationalizing the denominator transforms an irrational expression into a simplified rational form, enhancing clarity and usability in advanced applications.", "---", "### Final Tip", "Always verify the domain restrictions after rationalizing: since ( \sqrt{2x + 1} ) requires ( 2x + 1 \geq 0 ), the domain of ( h(x) ) begins at ( x \geq -\frac{1}{2} ), and the square root must remain real-valued.", "By mastering denominator rationalization—especially multiplying by the conjugate-like expression ( \sqrt{2x + 1} )—you strengthen your algebraic toolkit and prepare for complex functions in calculus and higher mathematics.", "---", "Keywords: rationalize denominator, eliminate square root in denominator, algebra technique, simplify rational function, math tutorial, h(x) rationalization, calculus preparation"]