Problem 86 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The result of the division is \(x^2 - 1\).

Step 1: Set Up Division

Divide \(3x^4 - 2x^2 - 5\) by \(3x^2 - 5\) using polynomial long division. Set it up similar to numerical long division.

Step 2: Divide Leading Terms

Divide the leading term of the dividend \(3x^4\) by the leading term of the divisor \(3x^2\). This gives \(x^2\).

Step 3: Multiply and Subtract

Multiply \(x^2\) by the entire divisor \(3x^2 - 5\) to get \(3x^4 - 5x^2\). Subtract this from the original dividend \(3x^4 - 2x^2 - 5\) to get a new dividend \(3x^2 - 5\).

Step 4: Repeat Division Process

Now divide the new leading term \(-3x^2\) by the leading term of the divisor \(3x^2\). This gives \(-1\).

Step 5: Multiply and Subtract Again

Multiply \(-1\) by the divisor \(3x^2 - 5\) to get \(-3x^2 + 5\), and subtract this from the new dividend \(3x^2 - 5\). This gives a remainder of 0.

Step 6: Write Final Result

The quotient from the division process is \(x^2 - 1\) with a remainder of 0. So, the final result of the division is \(x^2 - 1\).

Key Concepts

Dividing Polynomials

Dividing polynomials, similar to numerical division, involves breaking down a polynomial expression into simpler, manageable pieces. The process begins with arranging the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by) in the correct order.
For example, in the exercise, the dividend is set as \(3x^4 - 2x^2 - 5\), and the divisor is \(3x^2 - 5\).

Next, the division starts by dividing the leading term of the dividend by the leading term of the divisor. This gives you the first term of the quotient. Using this method, the leading terms \(3x^4\) and \(3x^2\) are divided to initially get \(x^2\).

Once you have this term, multiply it by the entire divisor and subtract it from the original polynomial. This yields a simpler polynomial or a new dividend.

• Keep repeating these steps until the degree of the remaining polynomial (remainder) is less than the degree of the divisor.
• The final set of terms form the complete quotient for the division.

This method continues until the dividend is fully divided, and either there remains a remainder or it turns to zero, indicating that the polynomial division is complete.

Polynomial Operations

Polynomial operations include basic computational skills such as addition, subtraction, multiplication, and division of polynomials. Each operation follows specific rules and can be systematically accomplished using algebraic strategies.

Polynomial division is the inverse process of polynomial multiplication. When performing polynomial long division, as detailed in the exercise, each operation must be carefully executed. The procedure is closely related to basic arithmetic long division, adapted to work with polynomial expressions.

• Addition and Subtraction: Combine like terms, which have the same variable raised to the same power.
• Multiplication: Distribute each term of the first polynomial to every term of the second polynomial.
• Division: As previously explored, involves breaking down the dividend by removing terms piece by piece.

Polynomial operations are fundamental skills used across various levels of algebra and calculus. Understanding the concepts enables students to solve complex polynomial equations and problems effectively.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and coefficients connected by operations such as addition, subtraction, multiplication, and division. Polynomials are a particular type of algebraic expression that includes terms with non-negative integer exponents.

In the context of polynomials, an algebraic expression can consist of multiple terms, like \(3x^4 - 2x^2 - 5\), with each term including:

• A coefficient, which is the numerical part (e.g., 3 or -2).
• A variable raised to a power, indicating the degree of the term (e.g., \(x^4\) or \(x^2\)).

Understanding algebraic expressions is crucial because they are foundational in solving equations, modeling real-world scenarios, and executing operations in algebra.

When dealing with algebraic expressions, it's essential to comprehend how they interact through operations. For instance, polynomial long division connects each piece of the expression by breaking it down, validating the knowledge of how variables and coefficients function together. Accurately manipulating these expressions builds a student's proficiency in algebra, from basic to more advanced levels.