Question

1. divide.

\\(\frac{2x^3 + 6x^2 + x + 4}{x}\\)
\\(\bigcirc\\) \\(2x^2 + 6x + \frac{4}{x}\\)
\\(\bigcirc\\) \\(2x^2 + 6x + 1 + \frac{4}{x}\\)
\\(\bigcirc\\) \\(x^2 + x + 1 + \frac{4}{x}\\)
\\(\bigcirc\\) \\(2x^2 + 6x + 4\\)

Explanation:

Step1: Divide each term by \( x \)

We have the expression \( \frac{2x^3 + 6x^2 + x + 4}{x} \). Using the rule of dividing a polynomial by a monomial \( \frac{a + b + c + d}{m}=\frac{a}{m}+\frac{b}{m}+\frac{c}{m}+\frac{d}{m} \), we divide each term in the numerator by \( x \).

For the first term: \( \frac{2x^3}{x}=2x^{3 - 1}=2x^2 \) (using the exponent rule \( \frac{x^m}{x^n}=x^{m - n} \) where \( m = 3\) and \( n=1\))

For the second term: \( \frac{6x^2}{x}=6x^{2 - 1}=6x \)

For the third term: \( \frac{x}{x}=1 \)

For the fourth term: \( \frac{4}{x} \) remains as it is since 4 and \( x \) have no common factors other than 1.

Step2: Combine the results

Combining all the divided terms, we get \( 2x^2+6x + 1+\frac{4}{x} \)

Answer:

B. \( 2x^2 + 6x + 1 + \frac{4}{x} \)