Question

question
simplify: \\(\frac{-6y^4 + 18y^3 + 12y^2 - 18y}{3y}\\)

Explanation:

Step1: Split the fraction

We can split the given fraction into four separate fractions by dividing each term in the numerator by the denominator \(3y\). So we have:
\(\frac{-6y^{4}}{3y}+\frac{18y^{3}}{3y}+\frac{12y^{2}}{3y}-\frac{18y}{3y}\)

Step2: Simplify each fraction

• For the first term \(\frac{-6y^{4}}{3y}\), we divide the coefficients and subtract the exponents of \(y\) (using the rule \(\frac{a^{m}}{a^{n}} = a^{m - n}\)). The coefficient \(-6\div3=-2\) and the exponent of \(y\) is \(4 - 1 = 3\), so it simplifies to \(-2y^{3}\).
• For the second term \(\frac{18y^{3}}{3y}\), the coefficient \(18\div3 = 6\) and the exponent of \(y\) is \(3-1=2\), so it simplifies to \(6y^{2}\).
• For the third term \(\frac{12y^{2}}{3y}\), the coefficient \(12\div3 = 4\) and the exponent of \(y\) is \(2 - 1=1\), so it simplifies to \(4y\).
• For the fourth term \(\frac{-18y}{3y}\), the coefficient \(-18\div3=-6\) and the exponent of \(y\) is \(1 - 1 = 0\) (and \(y^{0}=1\)), so it simplifies to \(-6\).

Answer:

\(-2y^{3}+6y^{2}+4y - 6\)