Problem 17 - Precalculus Mathematics for Calculus

Question

Find the quotient and remainder using long division. $$\frac{4 x^{3}+2 x^{2}-2 x-3}{2 x+1}$$

Step-by-Step Solution

Answer

The quotient is \(2x^2 - 1\) and the remainder is \(-2\).

Step 1: Set Up Long Division

To begin the long division, write the dividend \(4x^3 + 2x^2 - 2x - 3\) inside the division bracket and the divisor \(2x + 1\) outside. Align terms of the same degree vertically.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(2x\). This gives \(2x^2\) as the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the entire divisor \(2x + 1\) by the quotient term \(2x^2\) to get \(4x^3 + 2x^2\). Subtract this result from the original polynomial inside the division bracket to get \(0 + 0x^2 - 2x - 3\).

Step 4: Repeat Division

Next, take the new dividend \(-2x - 3\) and divide the first term \(-2x\) by the leading term of the divisor \(2x\), which gives \(-1\). This is the second term of the quotient.

Step 5: Multiply and Subtract Again

Multiply the divisor \(2x + 1\) by \(-1\) to get \(-2x - 1\). Subtract this from \(-2x - 3\), resulting in \(-2\). This is the remainder.

Step 6: Combine Quotient and Remainder

The quotient is \(2x^2 - 1\) and the remainder is \(-2\), therefore the result of the division is: \[ \frac{4x^3 + 2x^2 - 2x - 3}{2x + 1} = 2x^2 - 1 + \frac{-2}{2x + 1} \]

Key Concepts

Quotient and Remainder

When we perform polynomial long division, similar to long division in arithmetic, we find two important results: the quotient and the remainder. Suppose you have two polynomials and want to divide one by the other. The larger polynomial is called the dividend, and the smaller one is the divisor. The quotient is what you obtain from the division process, while the remainder is what is left over after the division.
In our specific problem, dividing the polynomial \( 4x^3 + 2x^2 - 2x - 3 \) by \( 2x + 1 \), the quotient is \( 2x^2 - 1 \). The remainder, in this case, is \(-2\). The final result of our division can be expressed as:

• Quotient: \( 2x^2 - 1 \)
• Remainder: \(-2\)

To write this, use the expression: \[ \frac{4x^3 + 2x^2 - 2x - 3}{2x+1} = 2x^2 - 1 + \frac{-2}{2x+1} \]

Dividing Polynomials

Dividing polynomials, much like dividing numbers, involves determining how many times one polynomial can "fit" into another. This process uses the concept of exact division, where we divide the leading terms first to simplify our work.
The basic idea is to align the polynomials, much like in long division with numbers, and divide the leading term of the dividend by the leading term of the divisor. In our example, we focus on the leading term \( 4x^3 \) from the dividend and \( 2x \) from the divisor. By dividing these terms, we begin building our quotient.

• Recognize the leading terms of both the dividend and the divisor.
• Divide these terms to find the first term of the quotient.

The remainder polynomial is then calculated and can be processed further or accepted as a remainder, based on whether it can be divided by the divisor further.

Polynomial Division Steps

To effectively perform polynomial division, follow a sequence of steps that simplify the process. These steps bring clarity and order, allowing for precise results.
1. **Set Up Long Division**: Write the dividend (\( 4x^3 + 2x^2 - 2x - 3 \)) inside the bracket and the divisor (\( 2x + 1 \)) outside. Make sure all terms are aligned by degree.2. **Divide Leading Terms**: Divide the leading term of the dividend (\( 4x^3 \)) by the leading term of the divisor (\( 2x \)) to get the first quotient term (\( 2x^2 \)).3. **Multiply and Subtract**: Multiply the entire divisor by this new term (\( 2x^2 \)) to achieve \( 4x^3 + 2x^2 \). Subtract this from the dividend to produce a new polynomial: \( 0 + 0x^2 - 2x - 3 \).

• Division and multiplication of terms help refine the solution at each step.

4. **Repeat if Necessary**: With the reduced polynomial \( -2x - 3 \), repeat the process. The first term here divided by \( 2x \) results in the next quotient term (\( -1 \)).5. **Conclude with Remainder**: After all possible divisions and subtractions, determine the final remainder. In our example, it's \(-2\), concluding the division.

• The methodical approach allows even complex expressions to be simplified step by step.