Problem 27 - Intermediate Algebra

Question

Perform each division. \(\frac{x^{2}+5 x+6}{x+3}\)

Step-by-Step Solution

Answer

The quotient is \(x + 2\).

Step 1: Set Up the Division

We start by setting up the polynomial long division. The polynomial to be divided is \(x^2 + 5x + 6\) and the divisor is \(x + 3\).

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(x^2\) by the leading term of the divisor \(x\). This gives \(x\).

Step 3: Multiply and Subtract

Multiply \(x\) by \(x + 3\) getting \(x^2 + 3x\). Subtract this from the original polynomial: \((x^2 + 5x + 6) - (x^2 + 3x) = 2x + 6\).

Step 4: Repeat Division

Divide the new leading term \(2x\) by the leading term of the divisor \(x\) to get \(2\).

Step 5: Multiply and Subtract Again

Multiply \(2\) by \(x + 3\) to get \(2x + 6\). Subtract this from \(2x + 6\) to get \(0\).

Step 6: Finalize the Answer

Combine the results from the divisions to get the quotient \(x + 2\). Since the remainder is zero, the division is exact.

Key Concepts

Long Division

Long division isn't just for numbers; it's a handy method to divide polynomials too. To understand this technique, let's see how it breaks down into steps. We start by arranging the terms of the dividend and the divisor in descending order of their degrees. Then, much like number division, we focus on the leading terms to guide us through each step.

• Divide: Look at the leading term of your dividend and see how many times it can be divided by the leading term of the divisor.
• Multiply: Multiply the entire divisor by the result you got from the division.
• Subtract: Subtract the result from the dividend. This step aligns closely with numerical long division where we subtract at each stage before bringing down the next term.
• Repeat: Bring down the next part of the polynomial and repeat until you've covered all terms.

Using this method offers a systematic approach to solving polynomial divisions and helps prevent errors that might otherwise occur in mental math.

Dividend and Divisor

In polynomial division, understanding the roles of the dividend and the divisor is crucial.
The dividend is the polynomial you're dividing into smaller parts. In our example, this is the polynomial \(x^2 + 5x + 6\). It's the polynomial that sits under the division bar in our setup.
The divisor, on the other hand, is the polynomial you're dividing by, often seen beside the division bar. For us, that's \(x+3\). The divisor determines the scale of the problem and what you need to "measure against" in each step.

• The coefficients in the dividend will change step by step as you subtract multiples of the divisor from them during the process.
• The divisor remains unchanged throughout the division process but guides you through each division step by examining its leading term.

Quotient and Remainder

The result of your polynomial division will yield two key components: the quotient and the remainder.
The quotient is what you get from dividing the leading terms of the dividend by the divisor and accumulating these values through each division step. It represents the simplified form of the division. In our example, the quotient is \(x + 2\).
When performing polynomial division, the goal is often to ensure that the remainder is as small as possible or zero.

• A remainder is what's left after the division process is complete. If the remainder is zero, like in our example, the division is considered exact or even.
• If there is a remainder, it reflects that the division isn’t exact, similar to how leftover parts work in regular number division.

Finally, it's this understanding of how the dividend breaks down into a quotient and a remainder that forms the core of polynomial division. By mastering these parts, you gain a comprehensive grasp of not just solving, but also interpreting polynomial divisions.