Problem 35 - College Algebra with Modeling and Visualization

Question

Use division to express the (Dividend) as (Divisor)(Quotient) \(+\) (Remainder) $$\frac{2 x^{3}+x^{2}-2 x}{2 x+1}$$

Step-by-Step Solution

Answer

The expression is \((2x + 1)(x^2 - 1) + 1\).

Step 1 - Set Up Long Division

To solve this problem, we will use polynomial long division. The dividend is \(2x^3 + x^2 - 2x\) and the divisor is \(2x + 1\). Write \(2x^3 + x^2 - 2x\) under the long division symbol and \(2x + 1\) outside.

Step 2 - Divide Leading Terms

Divide the leading term of the dividend, \(2x^3\), by the leading term of the divisor, \(2x\). This gives \(x^2\). Write \(x^2\) above the division line.

Step 3 - Multiply and Subtract

Multiply \(x^2\) by the divisor \(2x + 1\), which gives \(2x^3 + x^2\). Write this below \(2x^3 + x^2\) and subtract, which results in \(0x^3 + 0x^2 - 2x\) or simply \(-2x\).

Step 4 - Bring Down the Next Term

Since there is no constant term in the dividend, we don't have anything to bring down. The remaining expression is \(-2x\).

Step 5 - Repeat the Process

Divide the remaining term, \(-2x\), by the leading term of the divisor, \(2x\), to get \(-1\). Write \(-1\) above the division line. Multiply \(-1\) by \(2x + 1\), resulting in \(-2x - 1\). Write this below \(-2x\) and subtract, this results in a remainder of \(+1\).

Step 6 - Write the Final Expression

The quotient obtained is \(x^2 - 1\) and the remainder is \(+1\). Therefore, the expression of the dividend is \((2x + 1)(x^2 - 1) + 1\).

Key Concepts

Long Division

Polynomial long division is a process similar to the traditional long division method used with numbers, but it involves dividing polynomials. This technique helps us divide a polynomial, called the dividend, by another polynomial, the divisor. To begin, write the dividend under the long division symbol and the divisor outside, akin to regular long division. The goal is to determine how many times the leading term of the divisor fits into the leading term of the dividend.

The steps involve:

• Dividing the leading terms
• Multiplying back the result
• Subtracting to find the remainder
• Repeating the process with the new dividend

This step-by-step process continues until the remaining expression (remainder) cannot be divided further by the divisor.

In the example provided, we begin with the dividend \(2x^3 + x^2 - 2x\) and divisor \(2x + 1\). Writing these in the long division format sets the stage for our computation.

Dividend and Divisor

In polynomial division, understanding the roles of the dividend and the divisor is crucial. The dividend is the polynomial that you want to divide. For our example, the dividend is \(2x^3 + x^2 - 2x\). It's essentially the bigger polynomial from which we are aiming to isolate parts.

The divisor is the polynomial by which we divide the dividend. In the current exercise, the divisor is \(2x + 1\). It's usually a smaller polynomial, and its leading term is key in the initial steps of division.

These two elements are pivotal in setting up the division problem correctly. The process begins by focusing on the leading terms of both the dividend and divisor. Each step thereafter aims to reduce the complexity of the dividend by subtracting the product of the divisor and a computed multiplier from it.

Quotient and Remainder

The quotient and remainder result from the polynomial long division process. The quotient tells us how many times the divisor fits into the dividend completely. In this scenario, the quotient is \(x^2 - 1\). It is obtained from the successive division of the leading terms of the dividend and divisor as the process unfolds.

The remainder, on the other hand, is what's left over after completing the division. The remainder here is \(+1\). Unlike the quotient, the remainder should have a lower degree than the divisor. If the division can be perfectly executed with no leftovers, the remainder is zero.

When put together, the division of a polynomial is expressed as: \[(\text{Dividend}) = (\text{Divisor})(\text{Quotient}) + (\text{Remainder})\]This structured expression is crucial to verifying the accuracy of a polynomial division.

Polynomial Expressions

Polynomial expressions, fundamental elements of algebra, consist of variables raised to whole number powers and multiplied by coefficients. These mathematical constructs can appear simple, like \(x + 1\), or more complex, like \(2x^3 + x^2 - 2x\).

Polynomials are often used in functions, equations, and various mathematical expressions. They can be added, subtracted, multiplied, and as seen here, divided using specific techniques like polynomial long division.

Understanding how to manipulate polynomial expressions involves recognizing their structure:

• Coefficients: numerical factors of the terms
• Variables: symbols representing numbers
• Exponents: indicate the power to which the variable is raised

An in-depth grasp of these components aids in carrying out operations such as division successfully. This builds the foundation for solving more complex algebraic problems.