Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Use long division to divide. Divisor \(3 x^{2}-2\) Dividend $$3 x^{3}-12 x^{2}-2 x+8$$

Step-by-Step Solution

Answer

The result of this polynomial division is \(x - 4x\), with a remainder of \(-2x + 8\).

Step 1: Arrange divisor and dividend

Set up the division, putting the dividend \(3 x^{3}-12 x^{2}-2 x+8\) inside and \(3 x^{2}-2\) outside the division symbol.

Step 2: Divide the first terms

Divide the first term of the dividend, \(3x^3\), by the first term of the divisor, \(3x^2\). The result is \(x\) which we write above the division bar.

Step 3: Multiply and subtract

Now, multiply the divisor, \(3 x^{2}-2\), by \(x\) and subtract the result from the dividend. Write the result of this subtraction, \(-12x^2 - x + 8\), under the dividend by aligning like terms together.

Step 4: Repeat process

Repeat the prior steps with the new result as the current dividend. Divide \(-12x^2\) (the leading term of the new dividend) by \(3x^2\), to get \(-4x\). Multiply the original divisor, \(3 x^{2}-2\), by \(-4x\) and subtract this from the current dividend.

Step 5: Repeat again for the remainder

Keep repeating the process until you reach a remainder whose degree (the highest power of x in the polynomial) is less than the degree of the divisor. Divide \(-2x\) by \(3x^2\) and since the degree of \(-2x\) is less than \(3x^2\), we stop at this step.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, much like long division in arithmetic. In the context of the problem provided, the process involves dividing the dividend, which is the polynomial you're dividing, by the divisor. In this particular example, the dividend is the polynomial \(3x^3 - 12x^2 - 2x + 8\), and the divisor is \(3x^2 - 2\). The aim is to simplify the polynomial into a quotient, which is the result of the division, and possibly a remainder, which is what's left over if the polynomial does not divide evenly. To perform polynomial division, you need to:

• Set up the division as you would in long division, with the dividend under the division bar and the divisor outside.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this resulting term and subtract from the dividend.
• Repeat the process using the new polynomial that results from subtraction until the remainder is of a lower degree than the divisor.

This method is powerful for simplifying expressions and solving polynomial equations.

Algebraic Techniques

Algebraic techniques in polynomial division involve systematic strategies for simplifying complex algebraic expressions. The long division technique is one important algebraic method that allows you to breakdown a polynomial into simpler parts. To better understand these techniques, consider each step:

• Divide: Always start by dividing the first term of the current dividend by the first term of the divisor. This step helps you determine the next term in the quotient.
• Multiply: Take the term found in the division step and multiply the entire divisor by it. This multiplication gives you a new polynomial that you will subtract in the next step.
• Subtract: Subtract the result of the multiplication from the current dividend, which gives a new polynomial to repeat the process with.
• Repeat: Continue the cycle of divide, multiply, and subtract until the degree of the new polynomial, which becomes your remainder, is less than that of the divisor.

These steps are iterated upon until the termination condition is met. The quotient then collects all the terms found during the division steps, and any lower degree polynomial left is the remainder. These algebraic techniques are essential for solving the problem accurately and efficiently.

Remainders in Division

In polynomial division, the remainder plays a crucial role in understanding the limits of the division performed. The remainder is the polynomial that remains when the division process cannot proceed further with terms of higher degree. For the given example, when dividing \(-2x\) by \(3x^2\), the process stops as the degree of \(-2x\) is less than that of \(3x^2\). Here, the remainder provides valuable information:

• The degree of the remainder is always less than the degree of the divisor.
• If the remainder is zero, it means the dividend is exactly divisible by the divisor.
• If there is a non-zero remainder, like in this exercise, it indicates what is left over after the division.

In real-world applications, the remainder can sometimes represent an error in approximation or the part of the equation that is not accounted for in a clean division. Understanding how to interpret and work with remainders is vital for fully grasping the concept of polynomial division and applying it to various mathematical problems.