4) 7 x^3+x^2+x div x^2+1 Long division

To perform long division on the given polynomial \( 7x^3 + x^2 + x \) divided by \( x^2 + 1 \), we follow these steps:<br /><br />1. **Divide the leading term of the dividend by the leading term of the divisor:**<br /> \[<br /> \frac{7x^3}{x^2} = 7x<br /> \]<br /><br />2. **Multiply the entire divisor by this result and subtract from the original polynomial:**<br /> \[<br /> (7x)(x^2 + 1) = 7x^3 + 7x<br /> \]<br /> \[<br /> (7x^3 + x^2 + x) - (7x^3 + 7x) = x^2 - 6x<br /> \]<br /><br />3. **Repeat the process with the new polynomial \( x^2 - 6x \):**<br /> \[<br /> \frac{x^2}{x^2} = 1<br /> \]<br /> \[<br /> (1)(x^2 + 1) = x^2 + 1<br /> \]<br /> \[<br /> (x^2 - 6x) - (x^2 + 1) = -6x - 1<br /> \]<br /><br />4. **Since the degree of the remainder \(-6x - 1\) is less than the degree of the divisor \(x^2 + 1\), we stop here.**<br /><br />So, the quotient is \( 7x + 1 \) and the remainder is \(-6x - 1\).<br /><br />Thus, the result of the division is:<br />\[<br />7x + 1 + \frac{-6x - 1}{x^2 + 1}<br />\]<br /><br />Or, more neatly written:<br />\[<br />7x + 1 - \frac{6x + 1}{x^2 + 1}<br />\]