Problem 49 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient and remainder. $$ \frac{4 x^{3}-33}{x-2} $$

Step-by-Step Solution

Answer

The quotient is \(4x^2 + 8x + 16\) with a remainder of -1.

Step 1: Identify Coefficients

Identify the coefficients of the polynomial \(4x^3 - 33\). Since there are no \(x^2\) or \(x\) terms, the coefficients are \([4, 0, 0, -33]\).

Step 2: Write Down the Division Setup

In synthetic division, write down the root of the divisor \(x-2=0\), which is 2. Write it to the left of the synthetic division bracket. Then, write the coefficients \([4, 0, 0, -33]\) to the right.

Step 3: Bring Down the Leading Coefficient

Bring down the first coefficient, 4, to the bottom row as it is to begin the synthetic division process.

Step 4: Multiply and Add

Multiply the root (2) by the number below the line (4) and write the result (8) under the next coefficient (0). Add these numbers to get 8. Repeat this process: multiply 2 by 8 (results in 16) and add to next coefficient (0) to get 16. Then multiply 2 by 16 (results in 32) and add to the final coefficient (-33) to get -1.

Step 5: Determine the Quotient and Remainder

The numbers on the bottom row represent the coefficients of the quotient, starting from one degree less than the original polynomial. Thus, the quotient is \(4x^2 + 8x + 16\) and the remainder is -1.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, similar to long division with numbers. It helps break down a complex polynomial into simpler parts, making calculations more manageable. In many cases, synthetic division is used as a shortcut technique for dividing a polynomial by a binomial of the form \(x - c\). This approach involves dealing only with coefficients and is particularly useful when the divisor is linear.

• Simplifies complex polynomials
• Used for dividing by linear binomials
• Involves dealing with coefficients

Understanding polynomial division is key for simplifying expressions, factoring polynomials, and solving polynomial equations.

Quotient and Remainder

In the context of polynomial division, the quotient and remainder have similar roles as in numeric division. The quotient is the polynomial result from the division, while the remainder is what's left after the division is complete. When dividing the polynomial \(4x^3 - 33\) by \(x - 2\), the synthetic division process provides us with these two components.

The division results in a quotient of \(4x^2 + 8x + 16\) and a remainder of -1. This result can be verified through polynomial multiplication and addition:

• Quotient: \(4x^2 + 8x + 16\)
• Remainder: -1

Understanding the quotient and remainder helps in reconstructing and verifying the original polynomial equation.

Step-by-Step Division

The process of synthetic division is broken down into clear, sequential steps for simplicity. These steps involve setting up the problem, performing calculations, and interpreting results. Here’s how you perform synthetic division for the polynomial \(4x^3 - 33\):

1. **Identify the Coefficients**: For the polynomial \(4x^3 - 33\), write the coefficients as \([4, 0, 0, -33]\).
2. **Set Up the Division**: Use the root of the divisor \(x-2\), which is 2, and place it to the left of the coefficients.
3. **Begin the Calculation**: Bring down the first coefficient (4) directly under the line.
4. **Perform Multiply and Add Steps**: Multiply the root by the number below and add to the next coefficient iteratively.
5. **Interpret Results**: The bottom row gives the quotient and the last value is the remainder.

• Starts with coefficient identification
• Involves simple multiplication and addition
• Ends with reading the results for quotient and remainder

These steps make synthetic division fast and efficient, especially compared to traditional polynomial division.

Coefficient Identification

Coefficient identification is an essential first step in synthetic division, as it sets the stage for the entire process. Coefficients are the numerical factors in each term of a polynomial equation.

• Identify coefficients from equation
• Include 0 for missing degrees
• Arrange in a sequential list

For the polynomial \(4x^3 - 33\), the coefficients are \([4, 0, 0, -33]\), since there are no terms for \(x^2\) and \(x\). This arrangement is crucial because it aligns the calculation process with the powers of \(x\) in a systematic manner. Proper coefficient identification prevents errors and ensures accurate division outcomes.