Question

Express (4x3 + 5x + 1)/(x2 + 1) using long division method in the form q(x) + r(x)/b(x) where q(x) is the quotient, r(x) is the remainder, and b(x) is the divisor.

Answer 1

Answer:

4x + (x + 1) / (x^2 + 1)

Explanation:

We perform the long division

The result of the above long division tells is that

$4x^3+5x+1=4x(x^2+1)+(x+1)$

If we now divide both sides by x^2 + 1, we get

$\frac{4x^3+5x+1}{x^2+1}=\frac{4x(x^2+1)+(x+1)}{x^2+1}$$=\frac{4x(x^2+1)}{x^2+1}+\frac{(x+1)}{x^2+1}$$=4x+\frac{x+1}{x^2+1}$

Hence,

$\boxed{\frac{4x^3+5x+1}{x^2+1}=4x+\frac{x+1}{x^2+1}\text{.}}$

Therefore, the first choice from the options is the correct answer!