Problem 33 - Algebra 2

Question

Simplify. $$ \frac{2 x^{4}+3 x^{3}-2 x^{2}-3 x-6}{2 x+3} $$

Step-by-Step Solution

Answer

The simplified form is \(x^3 - x - 3\).

Step 1: Set up the Polynomial Long Division

Start by writing the dividend, which is the numerator, \(2x^4 + 3x^3 - 2x^2 - 3x - 6\), and the divisor, \(2x + 3\). We will divide the leading term of the dividend by the leading term of the divisor.

Step 2: Divide Leading Terms

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

Step 3: Multiply and Subtract

Multiply \(x^3\) by the entire divisor \(2x + 3\) to get \(2x^4 + 3x^3\). Subtract this product from the dividend to get a new polynomial \(0x^4 + 0x^3 - 2x^2 - 3x - 6\). Simplify to just \(-2x^2 - 3x - 6\).

Step 4: Repeat Division Process

Take the new polynomial \(-2x^2 - 3x - 6\) and divide the leading term \(-2x^2\) by the leading term of the divisor \(2x\), giving \(-x\).

Step 5: Multiply and Subtract Again

Multiply \(-x\) by \(2x + 3\) resulting in \(-2x^2 - 3x\). Subtract this from \(-2x^2 - 3x - 6\) to get \(0x^2 + 0x - 6\), which simplifies to \(-6\).

Step 6: Final Division

Finally, divide \(-6\) by the leading coefficient of \(2x\), which is just a constant term division, giving \(-3\).

Step 7: Construct and Verify the Quotient

The quotient from the division process is \(x^3 - x - 3\). Verify by multiplying the quotient \(x^3 - x - 3\) by the divisor \(2x + 3\) to ensure it equals the original dividend. The division had a remainder of zero.

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is the process of making them easier to work with by combining like terms or reducing them based on mathematical operations. When you simplify, you transform a complex expression into a simpler form without changing its value. This process might involve operations such as factoring, canceling, expanding, or polynomial long division, among others.

For example, consider the expression we are working with: \[\frac{2x^4 + 3x^3 - 2x^2 - 3x - 6}{2x + 3}\]By using polynomial long division, we simplify this complex fraction into the quotient:\[x^3 - x - 3\]
Simplification is often a crucial step in solving algebraic equations or inequalities, as it can reveal underlying properties or solutions that are easier to interpret or compute.

Algebraic Division

Algebraic division, especially polynomial long division, is a method that extends the logic of arithmetic division to algebraic expressions. This technique is particularly useful for dividing polynomials, where you divide the polynomial just like you would with numbers, using a systematic approach.

Here are the main steps to perform algebraic division:

• Divide: Start with the leading term of the dividend and divide it by the leading term of the divisor.
• Multiply: Multiply the entire divisor by the quotient obtained from the first step.
• Subtract: Subtract the result from the original dividend to obtain a new polynomial.
• Repeat: Continue this process with the remainder until the polynomial cannot be divided further.

Each of these steps helps break down the problem into manageable parts, allowing you to find the quotient and sometimes a remainder. In the given exercise, the remainder was zero, indicating that the division was exact, and the quotient was simply a simplified polynomial.

Polynomial Expressions

Polynomial expressions are mathematical expressions made up of variables and coefficients, constructed using only addition, subtraction, and multiplication. They are an essential component in algebra and calculus, as they can represent a wide variety of functions and equations.

A polynomial looks like this:\[ax^n + bx^{n-1} + \, \dots \, + k\]where each term consists of a coefficient (a numerical value), a variable (often \(x\)), and an exponent indicating the power of the variable. The highest exponent defines the degree of the polynomial, which indicates its complexity.

The polynomial presented in the exercise is a fourth-degree polynomial, as shown in the expression:\[2x^4 + 3x^3 - 2x^2 - 3x - 6\]Understanding polynomial expressions helps in the simplification process, as you know how each term behaves during operations like long division or when determining end behavior based on its degree. The ability to manipulate these expressions is vital for high-level algebra, calculus, and beyond.