In Exercises 27–29, divide using long division. $(4x^4+6x^3+3x-1)÷(2x^2+1)$

Verified step by step guidance

1. Write the division in long division format, placing the dividend \$4x^{4} + 6x^{3} + 3x - 1\( under the division bar and the divisor \)2x^{2} + 1$ outside.
2. Divide the leading term of the dividend \$4x^{4}\( by the leading term of the divisor \)2x^{2}\( to find the first term of the quotient: \)\frac{4x^{4}}{2x^{2}} = 2x^{2}$.
3. Multiply the entire divisor \$2x^{2} + 1\( by the term \)2x^{2}$ and subtract the result from the dividend to find the new remainder.
4. Bring down the next terms from the original dividend as needed, then repeat the division process: divide the leading term of the new remainder by \$2x^{2}$, multiply the divisor by this term, and subtract again.
5. Continue this process until the degree of the remainder is less than the degree of the divisor \$2x^{2} + 1$, then write the quotient and the remainder.

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 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 when to stop the division process, as the division ends when the remainder's degree is less than the divisor's degree.

Like Terms and Polynomial Subtraction

Combining like terms means adding or subtracting terms with the same variable raised to the same power. During polynomial long division, subtracting the product of the divisor and the current quotient term from the dividend requires careful alignment and combination of like terms to simplify the expression correctly.