Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x+\frac{1}{3}, \quad 3 x^{4}+x^{3}-3 x+1 $$

Step-by-Step Solution

Answer

No, \(x + \frac{1}{3}\) is not a factor of \(3x^4 + x^3 - 3x + 1\).

Step 1: Identify the Divisor

The divisor given is the expression \(x + \frac{1}{3}\). To use synthetic division, we need the opposite sign of the constant term, which is \(-\frac{1}{3}\). This is the value we will use in synthetic division.

Step 2: Arrange the Dividend Coefficients

The dividend is \(3x^4 + x^3 - 3x + 1\). Rewrite this by listing its coefficients: \([3, 1, 0, -3, 1]\). The coefficient for \(x^2\) is 0 because there is no \(x^2\) term in the dividend.

Step 3: Perform Synthetic Division

Using \(-\frac{1}{3}\), begin the synthetic division with the coefficients:1. Bring down the leading coefficient: 3.2. Multiply \(3\) by \(-\frac{1}{3}\), getting \(-1\). Add \(1\) to \(-1\), result: 0.3. Multiply 0 by \(-\frac{1}{3}\), getting 0. Add 0 to 0, result: 0.4. Multiply 0 by \(-\frac{1}{3}\), getting 0. Add \(-3\) to 0, result: \(-3\).5. Multiply \(-3\) by \(-\frac{1}{3}\), getting 1. Add 1 to 1, result: 2.

Step 4: Interpret the Remainder

The remainder from the synthetic division is 2. Since the remainder is not zero, \(x + \frac{1}{3}\) is not a factor of \(3x^4 + x^3 - 3x + 1\). If it had been a factor, the remainder would have been 0.

Key Concepts

Polynomial Division

Polynomial division is a process similar to numerical long division but applied to polynomial expressions. It allows breaking down a complex polynomial into simpler components. In this context, we are using synthetic division, a more straightforward method compared to long division, especially when dividing by a linear binomial like \(x + \frac{1}{3}\).
Synthetic division is specifically used because it reduces the complexity by utilizing only the coefficients of the polynomials.

• It is ideal for dividing by expressions of the form \(x - c\).
• Makes calculations quicker and less prone to error when handled correctly.

Understanding polynomial division through synthetic division helps with analyzing polynomials' behavior and solving them straight to their roots.
This method streamlines the division process, focusing on integer coefficients and leading easily to determining factors.

Factorization

Factorization involves expressing a polynomial as a product of its factors. It's like breaking a big problem into smaller, manageable pieces. In this scenario, synthetic division is employed to verify if \(x + \frac{1}{3}\) is a factor of the polynomial \(3x^4 + x^3 - 3x + 1\).
The factorization is dependent on the remainder after applying synthetic division. If no remainder is left, it confirms a successful factorization where:

• The divisor is indeed one of the factors.
• The polynomial can be expressed in terms of its factors, simplifying further calculations.

If a remainder appears, as in this exercise, it indicates that the divisor is not a factor. Thus, the polynomial's original form cannot be simplified using this divisor.

Coefficients

Coefficients are the numerical part of the terms in a polynomial. They play a crucial role in synthetic division as they are the numbers used in calculations.
In the polynomial \(3x^4 + x^3 - 3x + 1\), the coefficients are \([3, 1, 0, -3, 1]\). Note that a zero is included for the missing \(x^2\) term, ensuring all terms are represented in the process.

• Each coefficient directly impacts the outcome of synthetic division.
• Consistency in placing them correctly ensures accurate results.

By focusing on coefficients, synthetic division simplifies polynomial manipulation, allowing straightforward application without juggling algebraic terms. Understanding how coefficients interact is essential for mastering polynomial division.

Remainder Theorem

The remainder theorem is a fundamental principle when assessing whether a polynomial is divisible by a linear factor. It states that the remainder of a polynomial \(f(x)\) divided by \(x - c\) can be found by evaluating \(f(c)\).
In synthetic division, this translates to the final number in the division process being the remainder. When this remainder is zero, it confirms the polynomial is divisible by the factor, validating its presence as a factor.

• The remainder's value directly indicates the factorization possibility.
• A non-zero remainder means the divisor isn't a factor.

Understanding and applying the remainder theorem simplifies the process of verifying potential factors without the need for full polynomial division. This theorem is critical in algebraic contexts, especially in factorization and finding roots.