Problem 86 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The result is \(x^2\) with no remainder.

Step 1: Identify the Division

Recognize that the problem involves dividing the polynomial \(3x^4 - 2x^2 - 5\) by \(3x^2 - 5\). This means you need to use polynomial long division to solve the expression.

Step 2: Set Up the Division

Write the division in the long division format with \(3x^4 - 2x^2 - 5\) under the division symbol and \(3x^2 - 5\) outside.

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 \(x^2\). Write \(x^2\) above the division line.

Step 4: Multiply and Subtract

Multiply \(x^2\) by the divisor \(3x^2 - 5\), which gives \(3x^4 - 5x^2\). Subtract this result from \(3x^4 - 2x^2 - 5\) to get a new polynomial \(3x^4 - 2x^2 - 5 - (3x^4 - 5x^2) = 3x^2 - 5\).

Step 5: Bring Down Next Term

Since there are no more terms to bring down and the remainder \(3x^2 - 5\) is of higher degree than \(3x^2 - 5\), we stop here. The remainder is 0.

Step 6: Final Result

The final quotient is the term above the division line, which is \(x^2\), with a remainder of 0. Therefore, the result of the division is \(x^2\).

Key Concepts

Long Division

Long division is a method used in mathematics to divide one number or polynomial by another. Similar to numerical long division, polynomial long division involves dividing the highest terms first and systematically reducing the polynomial until you obtain the quotient and possibly a remainder. This concept is essential as it helps break down complex polynomial expressions into simpler components.

• Write the dividend and divisor in standard long division format.
• Focus on dividing the leading terms first—this gives you the first term of the quotient.
• Subtract the product of the divisor and the new quotient term from the current dividend.
• Repeat the process until the degree of the remainder is less than that of the divisor.

By mastering polynomial long division, solving complex algebraic expressions becomes more manageable, as seen in the division of polynomials like \( \frac{3x^4 - 2x^2 - 5}{3x^2 - 5} \).

Polynomial Functions

Polynomial functions are expressions consisting of terms in the form \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer. These functions are significant in algebra because they can represent a wide range of behaviors and can be used to model real-world phenomena such as trajectories, growth patterns, and trends.

• A polynomial's degree is determined by the highest exponent of \( x \). For example, \( 3x^4 - 2x^2 - 5 \) is a fourth-degree polynomial.
• The leading coefficient is the coefficient of the highest degree term, informing about the function’s end behavior.
• Polynomial terms are ordered from highest to lowest degree to simplify processes like division.

Understanding these characteristics prepares you for operations such as polynomial division, allowing you to simplify and solve polynomial equations effectively.

Algebraic Expressions

Algebraic expressions are combinations of variables, coefficients, and constants, linked by operations like addition, subtraction, multiplication, and division. They form the basis of algebra and are used to represent everything from simple equations to complex polynomial functions.

• Variables in algebraic expressions stand for numbers that can change or vary.
• Coefficients are the numerical factors multiplying the variables, indicating their strength or magnitude in the expression.
• Algebraic expressions can be simplified, expanded, and factored to reveal relationships between variables.

When learning about algebraic expressions, recognizing how they interact and how operations transform them is crucial for understanding polynomial division and other algebraic processes. This understanding helps tackle problems methodically, turning them into simpler equations or demonstrating fundamental principles.