Question

what is the quotient?
\frac{15x^2 - 10}{5x}
a. 3x^2 - 2
b. \frac{3x - 2}{x}
c. \frac{3x^3 - 2}{5}
d. \frac{3x^3 - 2}{x}

Explanation:

Step1: Split the fraction

We can split the fraction \(\frac{15x^{2}-10}{5x}\) into two separate fractions: \(\frac{15x^{2}}{5x}-\frac{10}{5x}\).

Step2: Simplify each fraction

For the first fraction \(\frac{15x^{2}}{5x}\), we divide the coefficients and subtract the exponents of \(x\) (using the rule \(\frac{x^{m}}{x^{n}} = x^{m - n}\)). So, \(\frac{15}{5}=3\) and \(\frac{x^{2}}{x}=x^{2 - 1}=x\), thus \(\frac{15x^{2}}{5x}=3x\).
For the second fraction \(\frac{10}{5x}\), we simplify the coefficient: \(\frac{10}{5}=2\), so \(\frac{10}{5x}=\frac{2}{x}\).
Putting it back together, we get \(3x-\frac{2}{x}\), which can be written as \(\frac{3x^{2}-2}{x}\) (by getting a common denominator \(x\) for \(3x\) and \(\frac{2}{x}\): \(3x=\frac{3x^{2}}{x}\), so \(\frac{3x^{2}}{x}-\frac{2}{x}=\frac{3x^{2}-2}{x}\))? Wait, no, wait. Wait, when we split \(\frac{15x^{2}-10}{5x}\) as \(\frac{15x^{2}}{5x}-\frac{10}{5x}\), simplifying \(\frac{15x^{2}}{5x}\) gives \(3x\) (since \(15\div5 = 3\) and \(x^{2}\div x=x\)), and \(\frac{10}{5x}=\frac{2}{x}\). So the expression is \(3x-\frac{2}{x}\), which is equivalent to \(\frac{3x^{2}-2}{x}\)? Wait, no, let's check the options. Option D: \(\frac{3x^{2}-2}{x}\)? Wait, the options:

Wait, the original problem's options: Let me re - check. The options are:

A. \(3x^{2}-2\)

B. \(\frac{3x - 2}{x}\)

C. \(\frac{3x^{3}-2}{5}\)

D. \(\frac{3x^{2}-2}{x}\)

Wait, when we simplify \(\frac{15x^{2}-10}{5x}\):

\(\frac{15x^{2}}{5x}-\frac{10}{5x}=\frac{15x^{2}}{5x}-\frac{10}{5x}\)

Simplify \(\frac{15x^{2}}{5x}\): \(15\div5 = 3\), \(x^{2}\div x=x\), so it's \(3x\).

Simplify \(\frac{10}{5x}\): \(10\div5 = 2\), so it's \(\frac{2}{x}\).

So \(3x-\frac{2}{x}=\frac{3x\times x-2}{x}=\frac{3x^{2}-2}{x}\), which is option D. Wait, but let's do the division correctly.

Alternative approach: We can factor the numerator? Wait, the numerator is \(15x^{2}-10 = 5(3x^{2}-2)\). Then \(\frac{5(3x^{2}-2)}{5x}=\frac{3x^{2}-2}{x}\) (we cancel out the common factor of 5 in the numerator and the denominator). Yes, that's a better way. Factor out 5 from the numerator: \(15x^{2}-10 = 5(3x^{2}-2)\). Then the fraction becomes \(\frac{5(3x^{2}-2)}{5x}\). The 5 in the numerator and denominator cancels out, leaving \(\frac{3x^{2}-2}{x}\), which is option D.

Answer:

D. \(\frac{3x^{2}-2}{x}\)