To perform the long division of polynomials to divide \( x^3 - x^2 + 4x + 2 \) by \( x \), we can follow these steps:
-
Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \), which gives us \( x^2 \).
-
Multiply the entire divisor \( x \) by \( x^2 \) to get \( x^3 \).
-
Subtract this result from the original polynomial:
\[ (x^3 - x^2 + 4x + 2) - (x^3) = -x^2 + 4x + 2 \]
-
Now repeat the process for the new polynomial \(-x^2 + 4x + 2\). Divide the leading term \(-x^2\) by \(x\), which gives \(-x\).
-
Multiply the divisor by \(-x\) to get \(-x^2\), then subtract:
\[ (-x^2 + 4x + 2) - (-x^2) = 4x + 2 \]
-
Now repeat the process for \(4x + 2\). Divide \(4x\) by \(x\), which gives us \(4\).
-
Multiply the divisor \(x\) by \(4\) to get \(4x\):
\[ (4x + 2) - (4x) = 2 \]
Now what we have is:
\[ x^2 - x + 4 \text{ with a remainder of } 2. \]
So we can write:
\[ \frac{x^3 - x^2 + 4x + 2}{x} = x^2 - x + 4 + \frac{2}{x} \]
Now let's match the response with the original options given:
The division result can be summarized as:
\[ x^2 - x + 4 + \frac{2}{x} \]
Here are the simplified terms put together for better clarity, and the correct response from the options you provided is:
Answer:
None of the options presented closely matches the division result of \( x^2 - x + 4 + \frac{2}{x} \).