Problem 40 - Basic Technical Mathematics with Calculus

Question

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$3 x^{4}-2 x^{3}+x^{2}+15 x+4 ; \quad 3 x+4$$

Step-by-Step Solution

Answer

Yes, \(3x + 4\) is a factor of the polynomial.

Step 1: Identify the Potential Root

The expression we need to check as a factor is \(3x + 4\). Using the Factor Theorem, we equate it to 0 to find the root: \(3x + 4 = 0\). Solving for \(x\), we get \(x = -\frac{4}{3}\). If \(3x + 4\) is a factor of the polynomial, then substituting \(x = -\frac{4}{3}\) into the polynomial should yield zero.

Step 2: Set Up Synthetic Division

We will use synthetic division to divide the polynomial \(3x^4 - 2x^3 + x^2 + 15x + 4\) by \(x + \frac{4}{3}\). Rewrite the divisor as \(x - (-\frac{4}{3})\). List the coefficients: \(3, -2, 1, 15, 4\).

Step 3: Perform Synthetic Division

Begin the synthetic division by bringing down the first coefficient (3) as is. Multiply the root \(-\frac{4}{3}\) by this number and add it to the next coefficient: - \((-\frac{4}{3}) \times 3 = -4; -2 + (-4) = -6\)Repeat the process for each coefficient to perform the complete division:- \((-\frac{4}{3}) \times -6 = 8; 1 + 8 = 9\)- \((-\frac{4}{3}) \times 9 = -12; 15 - 12 = 3\)- \((-\frac{4}{3}) \times 3 = -4; 4 - 4 = 0\)

Step 4: Analyze the Remainder

After performing the synthetic division, the last value is the remainder. Here, the remainder is 0, so the divisor \(x + \frac{4}{3}\) divides the polynomial evenly.

Key Concepts

Synthetic Division

Synthetic division is a streamlined method used to divide polynomials. It's particularly helpful when dividing by a simple linear expression, making calculations quicker and more efficient than traditional long division. In synthetic division, we focus solely on the coefficients of the polynomial, allowing us to manage calculations with ease.

Key Steps in Synthetic Division:

• Identify the potential root by setting the divisor equal to zero; for example, if your divisor is \(3x + 4\), solve for \(x = -\frac{4}{3}\).
• List the coefficients of the polynomial you are dividing. For the polynomial \(3x^4 - 2x^3 + x^2 + 15x + 4\), the coefficients are \[3, -2, 1, 15, 4\].
• Start the division by bringing down the first coefficient.
• Multiply the identified potential root by the coefficient just brought down, then add to the next coefficient. Continue this process across all coefficients.

This approach results in either a remainder or identifies the divisor as a true factor if the remainder is zero, as in our example.

Polynomial Division

Polynomial division, in general, is a process similar to the long division of numbers but applied to polynomials. It's essential for manipulating algebraic expressions, particularly when simplifying them or finding factors.

When to Use Polynomial Division:

• When dividing a polynomial by another polynomial, especially when examining potential factorization.
• Between any polynomial and a linear divisor like \(3x + 4\), which can be swiftly managed through synthetic division.

Polynomial division helps to simplify complex polynomials, identify roots, and demonstrate relationships between different algebraic expressions. It is a cornerstone of educational mathematics, offering foundational skills that add clarity and understanding to more advanced topics.

Educational Mathematics

Educational mathematics provides the framework to nurture critical thinking and problem-solving skills. Through exercises like polynomial division, students can better grasp mathematical concepts and techniques.

Why Use Exercises Like These:

• They illustrate abstract algebraic concepts in tangible terms, showing the step-by-step process from problem to solution.
• Help students understand the relationship between different mathematical methods, such as the Factor Theorem and synthetic division.
• Equip learners with the analytical skills necessary for higher-level mathematics and real-world applications.

These topics not only build knowledge but also encourage a structured approach to solving problems, essential for educational progress and confidence in mathematics.