Which of the following expressions is equivalent to [tex]$92x^2 + x - 1 + xy + \frac{2x^3}{x}$[/tex]?

A. [tex]$94x^2 + x + xy - 1$[/tex]
B. [tex]$92x^2 + 3x + xy - 1$[/tex]
C. [tex]$2x^2 + 92x + y - 1$[/tex]
D. [tex]$2x^3 + 92x^2 + 2x + xy - 1$[/tex]



Answer :

Certainly! Let's solve the problem step by step to determine which expression is equivalent to \(92 x^2 + x - 1 + xy + \frac{2 x^3}{x}\).

First, let's start by simplifying the given expression:

1. The original expression is:
[tex]\[ 92 x^2 + x - 1 + xy + \frac{2 x^3}{x} \][/tex]

2. Simplify \(\frac{2 x^3}{x}\):
[tex]\[ \frac{2 x^3}{x} = 2 x^2 \][/tex]

3. Substitute this simplified term back into the original expression:
[tex]\[ 92 x^2 + x - 1 + xy + 2 x^2 \][/tex]

4. Combine like terms \(92 x^2\) and \(2 x^2\):
[tex]\[ 92 x^2 + 2 x^2 = 94 x^2 \][/tex]

Therefore, the simplified expression is:
[tex]\[ 94 x^2 + x - 1 + xy \][/tex]

Now, let's look at each provided option to see which one matches \(94 x^2 + x - 1 + xy\) exactly:

- Option A \(94 x^2 + x + xy - 1\):
[tex]\[ 94 x^2 + x + xy - 1 \][/tex]
This matches our simplified expression exactly.

- Option B \(92 x^2 + 3 x + xy - 1\):
[tex]\[ 92 x^2 + 3 x + xy - 1 \][/tex]
This does not match our simplified expression. The \(92 x^2\) term and the \(3 x\) term are not equivalent to \(94 x^2\) and \(x\).

- Option C \(2 x^2 + 92 x + y - 1\):
[tex]\[ 2 x^2 + 92 x + y - 1 \][/tex]
This does not match our simplified expression. The terms are arranged differently, and their coefficients do not match \(94 x^2 + x - 1 + xy\).

- Option D \(2 x^3 + 92 x^2 + 2 x + xy - 1\):
[tex]\[ 2 x^3 + 92 x^2 + 2 x + xy - 1 \][/tex]
This does not match our simplified expression as it contains a \(2 x^3\) term, which is not in our simplified expression.

Therefore, the expression that is equivalent to \(92 x^2 + x - 1 + xy + \frac{2 x^3}{x}\) is:
[tex]\[ \boxed{A} \][/tex]