Problem 35 - Algebra and Trigonometry

Question

Find the quotient and remainder using synthetic division. \(\frac{x^{3}-27}{x-3}\)

Step-by-Step Solution

Answer

The quotient is \(x^2 + 3x + 9\) with a remainder of 0.

Step 1: Identify the terms

First, ensure the dividend, which is the polynomial being divided, is arranged in descending order of the exponents. In this case, the polynomial is already arranged as \(x^3 - 0x^2 + 0x - 27\). The divisor is \(x - 3\).

Step 2: Set up synthetic division

For synthetic division, we use the zero of the divisor \(x - 3\), which is \(3\). Write \(3\) on the left side and list the coefficients of the dividend polynomial on the right: \(1, 0, 0, -27\).

Step 3: Perform synthetic division

1. Bring down the first coefficient (1) directly below the line.2. Multiply \(3\) by this result (1) to get \(3\) and write it under the next coefficient (0).3. Add \(0 + 3 = 3\).4. Repeat: multiply \(3\) by this result (3) to get \(9\), write it under the next coefficient (0), then add \(0 + 9 = 9\).5. Repeat: multiply \(3\) by this result (9) to get \(27\), write it under the last coefficient (-27), then add \(-27 + 27 = 0\).

Step 4: Write the quotient and remainder

The numbers below the line are the coefficients of the quotient polynomial, while the last number is the remainder. Hence, the quotient is \(x^2 + 3x + 9\) with a remainder of \(0\).

Step 5: Final Verification

Since there's no remainder, we verify that the original polynomial \(x^3 - 27\) can be exactly divided by \(x - 3\), confirming our quotient \(x^2 + 3x + 9\) is correct.

Key Concepts

Polynomial Division

Polynomial division is a process used to simplify the division of one polynomial by another. This concept is similar to long division with numbers, but it uses polynomials. In our example, we are dividing the polynomial \(x^3 - 27\) by \(x - 3\).
In polynomial division, it is crucial to arrange the terms of the dividend in descending order of their exponents. This ensures that division is performed correctly from the term with the highest power to the term with the lowest power.
There are different methods to perform polynomial division, including long division and synthetic division. Synthetic division is often faster and consumes less space, especially for divisors of the form \(x - a\). It simplifies the process significantly, but it is limited to certain types of divisors.

Quotient and Remainder

When dividing polynomials, the result consists of a quotient and a remainder. The quotient is similar to what we get from traditional division, while the remainder is what is left over. For the example \(\frac{x^3 - 27}{x - 3}\), using synthetic division, we found:

• The quotient is \(x^2 + 3x + 9\).
• The remainder is \(0\).

Having a remainder of zero indicates that the polynomial divides evenly. The terms of the quotient identify the polynomial expression which, when multiplied by the divisor \(x - 3\), reconstructs the original polynomial \(x^3 - 27\).
In cases where there is a non-zero remainder, it indicates partial division, and the remainder is added to the quotient as a separate term \(\frac{remainder}{divisor}\).

Zero of the Divisor

The zero of the divisor is essential for synthetic division. It is the value that makes the divisor equal to zero. In the expression \(x - 3\), setting the divisor equal to zero gives \(x - 3 = 0\), therefore, \(x = 3\). This value, \(3\), is used in synthetic division.
Synthetic division uses this zero to simplify the computational process by only requiring the coefficients of the polynomial. It eliminates the need to rewrite variables during the calculation. By focusing only on the coefficients, we gain efficiency and simplicity. The zero provides the basis for the iterative multiplication and addition steps used to derive the quotient and remainder.

Division of Polynomials

Division of polynomials is a fundamental operation in algebra akin to dividing numbers. There are two primary ways to divide polynomials: long division and synthetic division.
Synthetic division is especially convenient for dividing polynomials by binomials of the form \(x - a\), like in the exercise given. It involves using the zero of the divisor and applying a series of multiplications and additions to reduce the polynomial.

• First, we write down the zero of the divisor.
• Second, only the coefficients of the polynomial are used.
• Third, each coefficient is processed, multiplied, and combined to derive the coefficients of the quotient.

This method simplifies the process and is much quicker compared to long division. It handles division without rewriting variable terms repeatedly, focusing just on numbers and basic arithmetic to reveal the quotient and remainder. This efficiency makes it a valuable tool, especially when dealing with larger polynomials or when speed is crucial.