Problem 19 - Algebra and Trigonometry

Question

Find the quotient and remainder using long division. \(\frac{6 x^{3}+2 x^{2}+22 x}{2 x^{2}+5}\)

Step-by-Step Solution

Answer

Quotient: \(3x + 1\), Remainder: \(2x\).

Step 1: Set Up the Long Division

Write the dividend \(6x^3 + 2x^2 + 22x\) and the divisor \(2x^2 + 5\) in a long division format. We will divide the terms starting with the highest degree term in the dividend.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(6x^3\) by the leading term of the divisor \(2x^2\). This gives us \(3x\). Write \(3x\) above the long division bar.

Step 3: Multiply and Subtract

Multiply the entire divisor \(2x^2 + 5\) by \(3x\) to get \(6x^3 + 15x\). Subtract \(6x^3 + 15x\) from \(6x^3 + 2x^2 + 22x\) to get \(2x^2 + 7x\).

Step 4: Repeat the Division Process

Now, consider \(2x^2 + 7x\). Divide the leading term \(2x^2\) by \(2x^2\) to get \(1\). Write \(1\) above the division bar next to \(3x\).

Step 5: Multiply and Subtract Again

Multiply \(1\) by the divisor \(2x^2 + 5\) to get \(2x^2 + 5\). Subtract from \(2x^2 + 7x\) to get the remainder \(2x\).

Step 6: Write the Final Answer

The quotient is \(3x + 1\) and the remainder is \(2x\). Therefore, the division can be expressed as:\[\frac{6x^3 + 2x^2 + 22x}{2x^2 + 5} = 3x + 1 + \frac{2x}{2x^2 + 5}\]

Key Concepts

Quotient and Remainder

When solving polynomial long division, two important results to identify are the quotient and the remainder. These results give us essential insight into how one polynomial interacts with another during division. The quotient represents the integer number resulting from division without considering the remainder. While on the other hand, the remainder is what is left over after the division process is completed. In our specific problem, dividing the polynomial \(6x^3 + 2x^2 + 22x\) by \(2x^2 + 5\) yields a quotient of \(3x + 1\) and a remainder of \(2x\).

With polynomial division, the calculation often looks like this:

• The problem is expressed as \( \frac{Dividend}{Divisor} = Quotient + \frac{Remainder}{Divisor} \)
• In our example substituting: \(\frac{6x^3 + 2x^2 + 22x}{2x^2 + 5} = 3x + 1 + \frac{2x}{2x^2 + 5}\)

The remainder is important because it tells us how much of the polynomial is still left undivided, illustrating the extent to which the dividend is evenly divided by the divisor.

Degree of Polynomial

In any polynomial, its degree is one of the most critical attributes. It signifies the highest power of the variable present in the polynomial. Understanding the degree of a polynomial helps us predict its behavior, especially during division tasks. The degree plays a crucial role in the process of polynomial long division because it determines how many terms the division will go through.

Let's break it down with our example:

• The degree of the dividend polynomial \(6x^3 + 2x^2 + 22x\) is 3, because of the term with \(x^3\).
• The degree of the divisor polynomial \(2x^2 + 5\) is 2, owing to the term with \(x^2\).

The degree of the quotient \(3x + 1\) is the result of subtracting the degree of the divisor from the degree of the dividend, in this case, \(3 - 2 = 1\). By understanding the degree, you can quickly assess how the division will proceed and whether the dividend can completely absorb the divisor by its corresponding terms.

Leading Term

The leading term of a polynomial is the term with the highest degree, and it's a pivotal element in conducting polynomial long division. We rely on leading terms because they guide us on which terms to divide and how to structure the quotient. During division, the primary action involves dividing the leading term of the dividend by the leading term of the divisor. This may sound complex, but it breaks down into simpler actions:

• Identify the leading term in the dividend, here it's \(6x^3\).
• Identify the leading term in the divisor, in our case it is \(2x^2\).
• Divide the leading term of the dividend by the leading term of the divisor: \(\frac{6x^3}{2x^2} = 3x\).

This calculation helps us find initial terms of the quotient like \(3x\). It is the backbone of the division process, displaying where the "action" happens in simplifying the expression. A solid understanding of leading terms can simplify tackling polynomial exercises, making the division process more intuitive.