Problem 344 - Precalculus

Question

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

Step-by-Step Solution

Answer

No, \(x + \frac{1}{3}\) is not a factor of the polynomial.

Step 1: Set Up Synthetic Division

The divisor is \(x + \frac{1}{3}\). To use synthetic division, we need the opposite of the constant term in the divisor, which is \(-\frac{1}{3}\). Write this on the left side. The coefficients of the polynomial 3x⁴ + x³ + 0x² - 3x + 1 are \(3, 1, 0, -3, 1\). Place these coefficients in a row.

Step 2: Perform Synthetic Division

Bring down the first coefficient (3) as it is. Multiply this by \(-\frac{1}{3}\) and add it to the next coefficient (1). Repeat this process for each coefficient:- \(3 \times -\frac{1}{3} = -1\), add to 1 to get 0.- \(0 \times -\frac{1}{3} = 0\), add to 0 to get 0.- \(0 \times -\frac{1}{3} = 0\), add to -3 to get -3.- \(-3 \times -\frac{1}{3} = 1\), add to 1 to get 2.

Step 3: Analyze the Remainder

The result of the synthetic division is \(3, 0, 0, -3, 2\). The last number (2) is the remainder. Since the remainder is not zero, \(x + \frac{1}{3}\) is not a factor of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is a process used to divide one polynomial by another, similar to how we divide numbers in arithmetic. The goal is to determine how many times the divisor fits into the dividend, forming a quotient. Synthetic division is a simplified form of polynomial division that is especially useful when dividing by linear polynomials of the form \(x - a\). This method streamlines calculations by focusing only on the coefficients of the polynomial. To set up synthetic division, it's essential to organize the coefficients of the dividend polynomials in descending order of power.

• First, determine the opposite of the constant term in the divisor for synthetic division.
• Next, list the coefficients from the highest degree to the lowest.
• Apply a series of multiplication and addition operations sequentially to simplify the division process.

This method helps avoid the complexity of handling variables explicitly, making it quicker and generally less error-prone for linear divisors.

Factors of Polynomials

Factors of polynomials are expressions that can be multiplied together to obtain the original polynomial. Recognizing factors is key to simplifying polynomial expressions and solving polynomial equations. When a divisor is a factor of the polynomial, dividing the polynomial by this factor will yield a zero remainder. This proof helps confirm the factorization.

• If the remainder is zero, then the divisor is indeed a factor.
• Expressing a polynomial as a product of simpler polynomials can reveal its root structure and solutions.
• Each factor corresponds to a potential root or solution of the polynomial equation.

Understanding factors helps in various applications, like solving equations or simplifying expressions, making complex problems more approachable.

Remainder Theorem

The Remainder Theorem provides a direct link between division and evaluating polynomials. According to this theorem, when a polynomial \(f(x)\) is divided by \(x - a\), the remainder of this division is the value \(f(a)\). This insight simplifies checking if \(x - a\) is a factor of the polynomial.

• The polynomial \(f(x)\) is divisible by \(x - a\) if and only if \(f(a) = 0\).
• If \(f(a) eq 0\), then \(f(a)\) is the remainder of the division.

In the context of synthetic division, the remainder theorem makes it easier to test for possible factors by evaluating the polynomial at specific values. This simplifies analysis greatly, especially for higher-degree polynomials, where traditional methods may involve cumbersome calculations.