Problem 136 - Introductory Algebra for College Students

Question

$$\text { Divide: } \frac{6 x^{2}+11 x-10}{3 x-2}$$

Step-by-Step Solution

Answer

\ The quotient is \(2x + 5\).

Step 1: Set Up The Long Division

Start by setting up the long division just as you would with numerical long division. Place the dividend (\(6x^2 + 11x - 10\)) inside the curve of the division bracket and the divisor (\(3x - 2\)) on the outside.

Step 2: Divide the First Terms

Divide the first term of the dividend (6x^2) by the first term of the divisor (3x). This gives \(2x\). Write this above the division bracket.

Step 3: Multiply and Subtract

Next, multiply the entire divisor (\(3x - 2\)) by the result from the previous step (\(2x\)), and write the product (\(6x^2 - 4x\)) under the dividend. Now, subtract this from the dividend. The operation will give you \(15x - 10\) which becomes the new dividend.

Step 4: Repeat the Process

Repeat the previous steps. Divide the first term of the new dividend (\(15x\)) by the first term of the divisor (\(3x\)), giving \(5\). Write this above the division bracket. Multiply the divisor by the new quotient and subtract from the new dividend to get a remainder of zero.

Step 5: Write Down the Answer

The result from the long division is the terms you wrote above the division bracket during the process. Thus, the division of \(6x^2 + 11x - 10\) by \(3x - 2\) equals \(2x + 5\).

Key Concepts

Long Division

Long division is a method used to divide one polynomial by another, similar to the way you perform long division with numbers. This method allows us to break down complex polynomials into more manageable pieces, making calculations easier and results clearer. Let's walk through how this process works when applied to polynomials:

• First, set up the long division by placing the dividend inside the division bracket, with the divisor outside.
• Next, divide the leading term of the dividend by the leading term of the divisor to determine the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend, similar to how subtraction is done during numerical division.
• Repeat this sequence of dividing, multiplying, and subtracting until you reach the end of the dividend or determine it's a polynomial remainder.

This method provides an organized way to handle complex terms, and ensuring each step is followed carefully will lead to the correct quotient and remainder.

Dividend

In the context of polynomial division, the dividend is the polynomial you wish to divide. It's the equivalent of the numerator in a fraction. In our initial example, the dividend is the polynomial \(6x^2 + 11x - 10\). This is the expression you place inside the division bracket.

• The dividend often determines the number of steps needed in the division process. You deal with it one term at a time, starting from the highest degree term.
• As you perform the division, you'll subtract parts of the dividend away, step by step, much like you would subtract multiples of the divisor in numerical division.

Dividing polynomials starts with handling the terms of the dividend, which makes understanding its structure crucial for successful division.

Divisor

The divisor is the polynomial by which you divide the dividend. In our example, \(3x - 2\) is the divisor. This polynomial determines how the terms of the dividend are reduced during each step of the division.

• The leading term of the divisor is especially important as it is used to divide each leading term of the dividend during the process.
• In each step, you multiply the divisor by the current term of the quotient and subtract the result from the current dividend to form a new dividend.

Understanding how the divisor interacts with the dividend is key to mastering polynomial long division.

Remainder

In polynomial division, the remainder is what remains after the division process is complete, when the dividend cannot be divided any further by the divisor. A remainder of zero, as seen in our example, indicates that the dividend is perfectly divisible by the divisor.

• The remainder is the portion left after subtracting the multiples of the divisor that have been handled throughout the division process.
• If a non-zero remainder exists, it means the original dividend was not fully divisible by the divisor using polynomial terms.

In our example, the process concluded with a remainder of zero, showing that \(6x^2 + 11x - 10\) divides evenly by \(3x - 2\), resulting in a neat polynomial quotient.