Perform each division. See Examples 9 and 10.
$\frac{(k^4-4k^2+2k+5)}{(k^2+1)}$

Verified step by step guidance

1. Identify the division problem as a polynomial division where you need to divide the numerator polynomial \(k^4 - 4k^2 + 2k + 5\) by the denominator polynomial \(k^2 + 1\).
2. Set up the long division by writing \(k^4 - 4k^2 + 2k + 5\) under the division bar and \(k^2 + 1\) outside the division bar.
3. Divide the leading term of the numerator, \(k^4\), by the leading term of the denominator, \(k^2\), to get the first term of the quotient, which is \(k^2\).
4. Multiply the entire divisor \(k^2 + 1\) by this quotient term \(k^2\) and subtract the result from the original numerator polynomial to find the new remainder polynomial.
5. Repeat the process: divide the leading term of the new remainder by \(k^2\), multiply the divisor by this new term, subtract again, and continue until the degree of the remainder is less than the degree of the divisor.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower or equal degree. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting from the dividend, and repeating until the remainder has a lower degree than the divisor.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree helps determine the steps in division and when to stop, as the division process continues until the remainder's degree is less than the divisor's degree.

Combining Like Terms

Combining like terms involves adding or subtracting terms with the same variable raised to the same power. This is essential during polynomial division to simplify expressions after subtraction and to correctly identify the new dividend for the next step.