Division of Polynomials
To find the quotient of \(\frac{5x^2 + 3x + 1}{2x}\), we will perform polynomial division.

Step 1: Arrange the Polynomial
First, we need to arrange the polynomial in descending order of exponents:
\(5x^2 + 3x + 1\)

Step 2: Divide the Leading Terms
Divide the leading term of the dividend by the leading term of the divisor:
\(\frac{5x^2}{2x} = \frac{5}{2}x\)

Step 3: Multiply and Subtract
Multiply the divisor by the result obtained in the previous step and subtract it from the dividend:
\(2x(\frac{5}{2}x) = 5x^2\)
\(5x^2 + 3x + 1 - 5x^2 = 3x + 1\)

Step 4: Repeat the Process
Now, we have \(3x + 1\) as the new dividend. Repeat steps 2 and 3.

Step 5: Divide the Leading Terms
\(\frac{3x}{2x} = \frac{3}{2}\)

Step 6: Multiply and Subtract
\(2x(\frac{3}{2}) = 3x\)
\(3x + 1 - 3x = 1\)

Step 7: Final Quotient
Since we have a constant term of 1 left, we can write the final quotient as:
\(\frac{5x^2 + 3x + 1}{2x} = \frac{5}{2}x + \frac{3}{2}\)
Therefore, the quotient of \(\frac{5x^2 + 3x + 1}{2x}\) when divided by 2x is \(\frac{5}{2}x + \frac{3}{2}\).

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