Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use long division to divide. $$\left(5 x-1+10 x^{3}-2 x^{2}\right) \div\left(2 x^{2}+1\right)$$

Step-by-Step Solution

Answer

The quotient is \(5x + 2.5\) and the remainder is \(2.5x - 1\).

Step 1: Arrange the Terms

Arrange the terms of both the dividend and the divisor in descending order of their degrees. This gives us \(10x^3 - 2x^2 + 5x - 1\) divided by \(2x^2 + 1\).

Step 2: Divide Leading Terms

Divide the leading term in the original dividend (which is \(10x^3\)) by the leading term in the divisor (which is \(2x^2\)). This gives us a first term in the quotient of \(5x\).

Step 3: Multiply and Subtract

Multiply the divisor \(2x^2 + 1\) by the term we just found (\(5x\)), subtract the result from the original dividend and bring down the next term. The difference becomes the next dividend, \(5x^2 + 5x - 1\).

Step 4: Continue the Process

Repeat the process. Divide the leading term of the new dividend by the leading term of the divisor. This gives a new term in the quotient, \(2.5x\). Multiply the divisor by this new term, subtract the result from the new dividend. The difference is \(2.5x - 1\).

Step 5: Final Step

Since \(2.5x - 1\) has a degree less than the degree of the divisor, we can not continue the division process. Thus, \(2.5x - 1\) is the remainder.

Key Concepts

Polynomial Division

Polynomial division is a crucial concept in algebra that lets us simplify complex expressions by dividing them into simpler parts. It works similarly to arithmetic long division, but instead of numbers, we're dealing with polynomials. Imagine you have a big mathematical cake and you want to cut it into smaller, more manageable pieces. That's what polynomial division does for algebraic expressions.
To perform polynomial division, you'll follow a series of steps to find both the quotient and the remainder. This involves dividing the leading terms, multiplying, and then subtracting until you reach a point where you can no longer divide. It's a systematic process that requires careful attention to detail with each step, similar to solving a puzzle.
Understanding polynomial division is not just about crunching numbers. It's about recognizing patterns and knowing how to manipulate expressions effectively. This skill is highly useful when you're dealing with complicated equations where simplifying terms is necessary.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of variables, coefficients, and operations such as addition, subtraction, multiplication, and division. Think of them as the ingredients in a recipe—each part is necessary and plays a crucial role in forming the final dish.
An expression like \(10x^3 - 2x^2 + 5x - 1\) might seem daunting at first. However, breaking it down into its components can simplify things. In this expression:

• \(10x^3\) is the term with the highest power, also known as the "leading term."
• \(-2x^2\) follows as the next highest power term.
• \(5x\) is the linear term, the simplest form of \(x\).
• \(-1\) is the constant term, which does not involve \(x\).

This breakdown helps make sense of the expression, showing how each part fits into the overall polynomial puzzle. Understanding these expressions is essential, especially when performing operations like addition, subtraction, or division.

Dividend and Divisor

In the context of polynomial division, the dividend and divisor are terms you'll frequently encounter. The dividend is the polynomial you want to divide, and the divisor is the polynomial you're dividing by.
For example, if you're working with \(10x^3 - 2x^2 + 5x - 1\) divided by \(2x^2 + 1\), here:

• \(10x^3 - 2x^2 + 5x - 1\) is the dividend.
• \(2x^2 + 1\) is the divisor.

Just like in numerical division, the goal is to see how many times the divisor fits into the dividend. By focusing on the leading terms, you determine how much of the divisor can "cover" the corresponding part of the dividend. This comparison is repeated until it is no longer feasible.
Understanding the relationship between the dividend and the divisor is vital in navigating through the steps of long division in algebra. This knowledge helps in calculating the quotient and identifying if there's any remainder left.

Remainder in Division

The remainder in division is what remains when you've divided as much as you can. In polynomial long division, it's the "leftover" part of the dividend that is smaller in degree than the divisor.
When dividing \(10x^3 - 2x^2 + 5x - 1\) by \(2x^2 + 1\), you perform the steps of dividing and subtracting, until you reach a polynomial degree lower than \(2x^2 + 1\). This results in a remainder of \(2.5x - 1\).
The remainder is crucial because it tells us how close our division came to being "even." When the remainder is zero, the divisor is a factor of the dividend, resulting in a perfect division. Otherwise, the remainder can provide insights into the nature of the division and help verify the calculations.
Recognizing and interpreting the remainder is a key part of mastering polynomial division, ensuring accurate solutions and deepening the understanding of algebraic processes.