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

Verified step by step guidance

1. Step 1: Set up the long division. Write the dividend \(4x^4 + 6x^3 + 3x - 1\) under the long division symbol and the divisor \(2x^2 + 1\) outside.
2. Step 2: Divide the first term of the dividend \(4x^4\) by the first term of the divisor \(2x^2\) to get the first term of the quotient, which is \(2x^2\).
3. Step 3: Multiply the entire divisor \(2x^2 + 1\) by the first term of the quotient \(2x^2\) and subtract the result from the original dividend.
4. Step 4: Bring down the next term from the dividend, if necessary, and repeat the process: divide the new leading term by the leading term of the divisor, multiply, and subtract.
5. Step 5: Continue this process until all terms of the dividend have been used, and write down the remainder, if any, after the last subtraction.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, and subtracting it from the dividend. This process is repeated with the new polynomial until the degree of the remainder is less than the degree of the divisor.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It determines the polynomial's behavior and the number of roots it can have. In the context of division, understanding the degrees of both the dividend and divisor is crucial, as it influences the number of times the division process will be performed and the form of the quotient.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is f(c). While this theorem specifically addresses linear divisors, it highlights the concept of remainders in polynomial division, which is also applicable in long division of polynomials. Understanding this theorem helps in verifying the results of polynomial division.