Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}-12 x^{3}+54 x^{2}-108 x+81\right) \div(x-3) $$

Step-by-Step Solution

Answer

The quotient is \(x^3 - 9x^2 + 27x - 27\).

Step 1: Verify Division Requirements

The divisor is already in the form \(x - c\) with \(c = 3\). Therefore, we can directly apply synthetic division without any need for adjustments.

Step 2: Set Up Synthetic Division

List the coefficients of the dividend \(x^4 - 12x^3 + 54x^2 - 108x + 81\), which are \([1, -12, 54, -108, 81]\). Use 3 (from \(x - 3\)) as the value to the side.

Step 3: Perform Synthetic Division

1. Bring down the leading coefficient (1) to the bottom row.2. Multiply by 3 and write the result under the next coefficient: \(3 \times 1 = 3\).3. Add the second coefficient: \(-12 + 3 = -9\).4. Multiply \(-9\) by 3: \(-9 \times 3 = -27\) and add to \(54\): \(54 - 27 = 27\).5. Multiply \(27\) by 3: \(27 \times 3 = 81\) and add to \(-108\): \(-108 + 81 = -27\).6. Multiply \(-27\) by 3: \(-27 \times 3 = -81\) and add to \(81\): \(81 - 81 = 0\).

Step 4: Write the Quotient

The bottom row of numbers \([1, -9, 27, -27, 0]\) represent the coefficients of the quotient. Drop the remainder (last number) since it is zero. The quotient is \(x^3 - 9x^2 + 27x - 27\).

Key Concepts

Polynomial Division

Polynomial division is a process similar to dividing numbers, but instead, we work with expressions that include variables with exponents. In this context, our goal is to divide a polynomial (the dividend) by another polynomial (the divisor), to find a quotient and possibly a remainder. This is especially useful in simplifying complex expressions and finding factors.
Imagine you're splitting a whole chocolate bar (polynomial) into equal parts. Each part here represents a term in the quotient with its specific degree based on where it fits in the division process. We follow steps much like numerical division:

• Align the terms based on their degree in descending order.
• Divide the leading terms to find the first term of the quotient.
• Multiply back, subtract from dividend, and repeat the process for the next term.

With synthetic division, we streamline this process significantly when the divisor is in the form of a linear binomial like \((x - c)\). This exercise, \(x^4 - 12 x^3 + 54 x^2 - 108 x + 81\) divided by \((x - 3)\), efficiently shows how concise the steps can be, giving us the quotient, \(x^3 - 9x^2 + 27x - 27\). This simplifies the initial polynomial without the remainder because it divides evenly.

Algebra Techniques

In algebra, various techniques are employed to solve equations or simplify expressions, and one such technique is synthetic division. It is a streamlined approach that aids in polynomial division specifically when the divisor is a linear expression. This method reduces the complexity by concentrating on coefficients rather than the variable terms.
Here's why synthetic division is a favored algebra technique:

• It only requires the coefficients of the polynomial, making calculations faster and less prone to error.
• Works seamlessly with linear divisors, often used in conjunction with the Remainder Theorem and Factor Theorem.

Using the exercise as a guide, you notice how each term results from multiplying and adding strategically, all without overarching algebraic clutter. This fosters a deeper understanding of the underlying algebraic relationships, preparing students for more advanced topics.

Division Algorithm

The division algorithm in algebra is a fundamental rule ensuring division results in a quotient and a remainder - a concept derived from numerical division. For polynomials, this dictum states that given any dividend \(f(x)\) and a non-zero divisor \(d(x)\), there exist unique polynomials (quotient \(q(x)\) and remainder \(r(x)\)) such that:\[ f(x) = d(x) \cdot q(x) + r(x) \]where the degree of \(r(x)\) is less than the degree of \(d(x)\).
This principle validates the steps in synthetic division, confirming that the output is both correct and consistent. In our given exercise, using \((x - 3)\) as a divisor means that the remainder is zero, affirming that \(x - 3\) is a factor of the polynomial \(x^4 - 12 x^3 + 54 x^2 - 108 x + 81\).
Understanding the division algorithm opens up possibilities in factoring larger polynomials and solving polynomial equations effortlessly. It lays the groundwork for finding roots and simplifying polynomial expressions using minimal calculation.