Perform each division. See Examples 9 and 10. $(x^4+5x^2+5x+27)/(x^2+3)$

Verified step by step guidance

1. Identify the division problem as a polynomial division: divide the polynomial \(x^4 + 5x^2 + 5x + 27\) by the polynomial \(x^2 + 3\).
2. Set up the long division by writing \(x^4 + 5x^2 + 5x + 27\) under the division bar and \(x^2 + 3\) outside the division bar.
3. Divide the leading term of the dividend \(x^4\) by the leading term of the divisor \(x^2\) to get the first term of the quotient: \(x^2\).
4. Multiply the entire divisor \(x^2 + 3\) by \(x^2\) and subtract the result from the dividend to find the new remainder.
5. Repeat the process: divide the leading term of the new remainder by \(x^2\), multiply the divisor by this 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 degree, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting, 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.

Remainder and Quotient in Polynomial Division

In polynomial division, the quotient is the result of the division, and the remainder is what is left when the division cannot continue. The remainder must have a degree less than the divisor, and the original polynomial can be expressed as divisor × quotient + remainder.