Question

the quotient of a polynomial and a monomial is written as $\frac{x^{4}+3x^{2}+\frac{3}{2}x}{\frac{1}{2}x}$
what is the value of the quotient?
a. $2x^{3}+6x+3$
b. $\frac{1}{2}x^{3}+\frac{3}{2}x+\frac{3}{4}$
c. $2x^{4}+6x^{2}+3x$
d. $\frac{1}{2}x^{4}+\frac{3}{2}x^{2}+\frac{3}{4}x$

Explanation:

Step1: Split the rational expression

$$\frac{x^4+3x^2+\frac{3}{2}x}{\frac{1}{2}x} = \frac{x^4}{\frac{1}{2}x} + \frac{3x^2}{\frac{1}{2}x} + \frac{\frac{3}{2}x}{\frac{1}{2}x}$$

Step2: Simplify each term

First term: $\frac{x^4}{\frac{1}{2}x} = 2x^{4-1}=2x^3$
Second term: $\frac{3x^2}{\frac{1}{2}x} = 6x^{2-1}=6x$
Third term: $\frac{\frac{3}{2}x}{\frac{1}{2}x} = 3$

Step3: Combine simplified terms

$$2x^3 + 6x + 3$$

Answer:

A. $2x^3 + 6x + 3$