Perform each division. See Examples 9 and 10. (k^4-4k^2+2k+5)/(k^2+1)

Verified step by step guidance

1. Identify the dividend and the divisor: The dividend is \(k^4 - 4k^2 + 2k + 5\) and the divisor is \(k^2 + 1\).
2. Set up the division: Write \(k^4 - 4k^2 + 2k + 5\) under the long division symbol and \(k^2 + 1\) outside.
3. Divide the first term of the dividend by the first term of the divisor: \(k^4 \div k^2 = k^2\). This is the first term of the quotient.
4. Multiply the entire divisor \(k^2 + 1\) by the first term of the quotient \(k^2\) and subtract the result from the dividend.
5. Repeat the process: Bring down the next term from the dividend, divide the new leading term by \(k^2\), multiply, and subtract until all terms are processed.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It involves determining how many times the divisor can fit into the leading term of the dividend, subtracting the result, and repeating the process with the remainder until the degree of the remainder is less than that of the divisor.

Synthetic Division

Synthetic division is a simplified form of polynomial division that is particularly useful when dividing by linear factors. It streamlines the process by using only the coefficients of the polynomials, making calculations quicker and easier, especially when the divisor is in the form of (k - c).

Remainder Theorem

The Remainder Theorem states that when a polynomial f(k) is divided by (k - c), the remainder of this division is equal to f(c). This theorem is useful for quickly finding the remainder without performing full polynomial division, and it can help in evaluating polynomials at specific points.