Question

Use long division to find the quotient and remainder of (x^4+3x^3-x^2+5x-3)/(x^2+1). Write the result as dividend = (divisor)(quotient)+remainder. What is the quotient in this division?

a) x^2
b) 3x
c) -1
d) 0

Answer 1

The quotient of the division obtained through long division is
`x^2`. Written in the form dividend = (divisor)(quotient) + remainder, the quotient corresponds to option a)
`x^2`.

Step-by-step explanation:

To find the quotient and remainder of the division
`(x^4+3x^3-x^2+5x-3)/(x^2+1),` we use long division. Here is the step-by-step process:

1. Divide the first term of the dividend
   `(x^4)` by the first term of the divisor
   `(x^2)` to get the first term of the quotient, which is
   `x^2`.
2. Multiply the divisor by
   `x^2` to get
   `x^4 + x^2` and subtract this from the dividend.
3. Bring down the next term of the original dividend to get a new dividend of
   `3x^3 - 2x^2` and repeat the process
   `3x`s.
4. Divide
   `3x^3` by
   `x^2`to get 3x, and repeat the multiplication and subtraction.
5. The process continues until the degree of the remainder is less than the degree of the divisor.

After completing the long division, you would have obtained a quotient, which is part of the answer. For the given question, the correct answer is the quotient
`x^2`, which is option a).

The written form of the division result is: dividend = (divisor)(quotient) + remainder.