Problem 28 - College Algebra with Modeling and Visualization

Question

Divide the expression. $$\frac{3 x^{4}-2 x^{2}-5}{3 x^{2}-5}$$

Step-by-Step Solution

Answer

The quotient is \( x^2 + 1 \) with no remainder.

Step 1: Recognize the Division

The given problem is an algebraic division of polynomials: \( \frac{3x^4 - 2x^2 - 5}{3x^2 - 5} \). This requires us to divide one polynomial by another, similar to long division with numbers.

Step 2: Set Up the Polynomial Division

Set up the division by writing the dividend \( 3x^4 - 2x^2 - 5 \) inside the division symbol and the divisor \( 3x^2 - 5 \) outside. Check that both are arranged in descending order of powers.

Step 3: Divide the Leading Terms

Divide the leading term of the dividend \( 3x^4 \) by the leading term of the divisor \( 3x^2 \). The result is \( x^2 \). This becomes the first term of the quotient.

Step 4: Multiply and Subtract

Multiply \( x^2 \) by the entire divisor \( 3x^2 - 5 \), which gives \( 3x^4 - 5x^2 \). Subtract this from the original dividend: \( (3x^4 - 2x^2 - 5) - (3x^4 - 5x^2) = 3x^2 - 5 \).

Step 5: Repeat the Process

Continue the process with the new dividend \( 3x^2 - 5 \). Divide \( 3x^2 \) by \( 3x^2 \) to get \( 1 \). Multiply this by \( 3x^2 - 5 \) to get \( 3x^2 - 5 \), and subtract again.

Step 6: Check for Remainder

After subtracting, the remainder is \( 0 \), indicating the division is exact and the polynomial divides evenly.

Key Concepts

Algebraic Division

Algebraic division is a method used to divide one algebraic expression by another. It is commonly used in algebra, especially when dealing with polynomials. Think of it as a way to "unbundle" a complex expression into simpler parts. Here are the steps to understand and apply algebraic division:

• Identify the dividend and the divisor in the algebraic expression. In the example given, the dividend is \(3x^4 - 2x^2 - 5\) and the divisor is \(3x^2 - 5\).
• The goal is to see how many times the divisor fits into the dividend. This is similar to asking how many times a number fits into another in basic arithmetic division.
• Transform algebraic expressions and terms to find out how each part of the divisor can be multiplied to achieve the dividend.

This method requires a strong understanding of polynomials and operations like addition, subtraction, multiplication, and division of terms.

Polynomial Long Division

Polynomial long division is a detailed method similar to the long division used with numbers. It is essential for dividing polynomials systematically. Let's explore how it functions:

• Set Up the Division: Organize the dividend and divisor in descending order of their powers before starting.
• Divide the Leading Terms: Focus on the leading term of each polynomial. In our problem, divide \(3x^4\) by \(3x^2\) to get \(x^2\). This is the quotient's first term.
• Multiply and Subtract: Multiply the quotient by the divisor and subtract the result from the original dividend. This adjusts the dividend for further steps.
• Repeat the Process: Use the new polynomial obtained from subtraction to repeat the process. Each step reduces the degree (power) of the remaining polynomial until there are no more terms of the same or higher degree than the divisor.

This process is repeated until a remainder is found (if any), allowing us to see how the dividend is constructed by the divisor parts.

Algebraic Expressions

Algebraic expressions are constructed using constants, variables, and arithmetic operations (addition, subtraction, multiplication, division). In the exercise, expressions are represented as polynomials, which are sums of terms and include variables with non-negative integer exponents.

• Understanding Terms: Each part of a polynomial is a term, like \(3x^4\), which consists of a coefficient \(3\) and a variable \(x\) raised to a power \(4\).
• Polynomials: These are particular types of algebraic expressions which have two or more terms. The expression \(3x^4 - 2x^2 - 5\) is a polynomial of degree 4, since the highest power of \(x\) is \(4\).
• Simplifying Expressions: Through operations such as polynomial division, complex expressions are simplified to reveal underlying relationships and structures.

Algebraic expressions facilitate modeling real-world scenarios and solving equations by breaking down components into manageable parts.