Question

divide the polynomial: \\(\frac{12x^3 - 18x^2}{6x}\\) \\(2x^2 - 3x\\) \\(2x^2 + 3x\\) \\(2x - 3x^2\\) \\(2x^2 - 3\\)

Explanation:

Step1: Split the fraction

We can split the given fraction \(\frac{12x^{3}-18x^{2}}{6x}\) into two separate fractions: \(\frac{12x^{3}}{6x}-\frac{18x^{2}}{6x}\).

Step2: Simplify each fraction

For the first fraction \(\frac{12x^{3}}{6x}\), we divide the coefficients and subtract the exponents of \(x\) (using the rule \(\frac{x^{m}}{x^{n}} = x^{m - n}\)). So, \(\frac{12}{6}=2\) and \(x^{3-1}=x^{2}\), giving \(2x^{2}\).
For the second fraction \(\frac{18x^{2}}{6x}\), \(\frac{18}{6} = 3\) and \(x^{2-1}=x\), so it simplifies to \(3x\). But since it's a subtraction, we have \(- 3x\).
Combining these two results, we get \(2x^{2}-3x\).

Answer:

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