Questions: The rational expression (2x^4-5x^2+3x-2)/(x-3) is equivalent to (1) 2x^3-5x-12-(38)/(x-3) (2) 2x^3+6x^2+13x+42+(124)/(x-3) (3) 2x^3-5x+18-(56)/(x-3) (4) 2x^3-6x^2+13x-36+(106)/(x-3)

Solution Steps

We start with the rational expression

\[ \frac{2x^{4}-5x^{2}+3x-2}{x-3}. \]

We perform polynomial long division of \(2x^{4}-5x^{2}+3x-2\) by \(x-3\).

1. Divide \(2x^{4}\) by \(x\) to get \(2x^{3}\).
2. Multiply \(2x^{3}\) by \(x-3\) to get \(2x^{4}-6x^{3}\).
3. Subtract this from the original polynomial, resulting in \(6x^{3}-5x^{2}+3x-2\).

Continuing this process:

• Divide \(6x^{3}\) by \(x\) to get \(6x^{2}\).
• Multiply \(6x^{2}\) by \(x-3\) to get \(6x^{3}-18x^{2}\).
• Subtract to get \(13x^{2}+3x-2\).

Next:

• Divide \(13x^{2}\) by \(x\) to get \(13x\).
• Multiply \(13x\) by \(x-3\) to get \(13x^{2}-39x\).
• Subtract to get \(42x-2\).

Finally:

• Divide \(42x\) by \(x\) to get \(42\).
• Multiply \(42\) by \(x-3\) to get \(42x-126\).
• Subtract to get a remainder of \(124\).

Thus, the quotient is

\[ 2x^{3} + 6x^{2} + 13x + 42 \]

and the remainder is \(124\).

Combining the quotient and the remainder, we express the original rational expression as:

\[ \frac{2x^{4}-5x^{2}+3x-2}{x-3} = 2x^{3} + 6x^{2} + 13x + 42 + \frac{124}{x-3}. \]

Final Answer