Polynomial Division A Step By Step Guide

Polynomial division, guys, might sound intimidating, but trust me, it's totally manageable once you break it down. In this guide, we're going to tackle the polynomial division problem ${\frac{2x^3 - 13x^2 + 9x - 16}{x - 6}}$. We'll explore the process step-by-step, ensuring you understand not just how to do it, but why it works. By the end, you'll be able to confidently divide polynomials and express your answers in the required form, which is either ${p(x)}$ or ${p(x) + \frac{k}{x - 6}}$, where ${p(x)}$ is a polynomial and ${k}$ is an integer. So, let's dive in and conquer this mathematical challenge together!

Understanding Polynomial Division

Polynomial division is essentially the algebraic version of long division you learned back in grade school. Remember dividing numbers like 125 by 5? We're doing the same thing here, but with polynomials instead of numbers. The goal is to divide the dividend (the polynomial being divided, in our case ${2x^3 - 13x^2 + 9x - 16}$) by the divisor (the polynomial we're dividing by, which is ${x - 6}$). The result will be the quotient (the polynomial result of the division) and possibly a remainder (the leftover part that doesn't divide evenly).

Think of it like this: if you have a large polynomial representing a complex shape's area, dividing it by a simpler polynomial (like the length of one side) might give you the polynomial representing the other side (the quotient). If there's a remainder, it's like having a small piece left over after perfectly fitting the shapes together.

Setting Up the Problem

Before we start, let's set up our polynomial division problem. We'll use a format similar to long division. Write the dividend (${2x^3 - 13x^2 + 9x - 16}$) inside the "division bracket" and the divisor (${x - 6}$) outside. Make sure the polynomials are written in descending order of their exponents (highest power of ${x}$ first). This helps keep things organized and prevents errors. Also, it's crucial to include placeholder terms with a coefficient of 0 for any missing powers of ${x}$. In our case, we have all the powers from ${x^3}$ down to the constant term, so we don't need placeholders.

The Division Process: A Step-by-Step Guide

Now, let's get to the heart of the matter: the actual division! Here's the step-by-step process we'll follow:

1. Divide the leading terms: Focus on the leading term of the dividend (${2x^3}$) and the leading term of the divisor (${x}$). Divide the first term of the dividend by the first term of the divisor. What do you get when you divide ${2x^3}$ by ${x}$? It's ${2x^2}$. This is the first term of our quotient.
2. Multiply the quotient term by the divisor: Multiply the ${2x^2}$ (the first term of our quotient) by the entire divisor (${x - 6}$). This gives us ${2x^2 * (x - 6) = 2x^3 - 12x^2}$. Guys, make sure you distribute the multiplication correctly!
3. Subtract: Write the result (${2x^3 - 12x^2}$) under the dividend and subtract. Pay close attention to the signs! We have ${(2x^3 - 13x^2) - (2x^3 - 12x^2) = -x^2}$.
4. Bring down the next term: Bring down the next term from the dividend (which is ${+9x}$) and write it next to the result of the subtraction. Now we have ${-x^2 + 9x}$.
5. Repeat: Repeat steps 1-4 with the new expression (${-x^2 + 9x}$).
   • Divide the leading term (${-x^2}$) by the leading term of the divisor (${x}$). This gives us ${-x}$, the next term of our quotient.
   • Multiply ${-x}$ by the divisor (${x - 6}$): ${-x * (x - 6) = -x^2 + 6x}$.
   • Subtract: ${(-x^2 + 9x) - (-x^2 + 6x) = 3x}$.
   • Bring down the next term (${-16}$): ${3x - 16}$.
6. Repeat Again: Repeat the process one more time.
   • Divide ${3x}$ by ${x}$, which gives us ${3}$, the last term of our quotient.
   • Multiply ${3}$ by the divisor: ${3 * (x - 6) = 3x - 18}$.
   • Subtract: ${(3x - 16) - (3x - 18) = 2}$.
7. The Remainder: We've reached a point where the degree of the remaining expression (which is just the constant ${2}$) is less than the degree of the divisor (which is ${x - 6}$). This means we've found our remainder. The remainder is ${2}$.

Expressing the Answer

Okay, so we've done the hard work! Now, how do we write the final answer? Remember, the form we're aiming for is either ${p(x)}$ or ${p(x) + \frac{k}{x - 6}}$.

Our quotient (the polynomial part) is ${2x^2 - x + 3}$. Our remainder is ${2}$. To express the remainder as a fraction, we put it over the divisor: ${\frac{2}{x - 6}}$.

Therefore, the final answer is:

${2x^2 - x + 3 + \frac{2}{x - 6}}$

Common Mistakes to Avoid in Polynomial Division

Polynomial division, like any mathematical process, is prone to errors if you're not careful. Here are some common pitfalls to watch out for:

• Sign Errors: Pay very close attention to signs, especially when subtracting polynomials. A single sign mistake can throw off the entire solution. Use parentheses to keep track of the negative signs during subtraction.
• Forgetting Placeholders: If a polynomial is missing a term (e.g., no ${x}$ term), make sure to include a placeholder with a coefficient of 0 (e.g., ${+0x}$). This helps maintain the correct alignment and prevents errors in the division process.
• Incorrect Multiplication: Make sure to distribute the multiplication correctly when multiplying the quotient term by the divisor. Multiply each term of the divisor by the quotient term.
• Stopping Too Early: Continue the division process until the degree of the remainder is less than the degree of the divisor. Don't stop prematurely!
• Misunderstanding the Format: Remember the required format for the answer: ${p(x)}$ or ${p(x) + \frac{k}{x - 6}}$. Make sure you express the remainder correctly as a fraction over the divisor.

Practice Makes Perfect: Examples and Exercises

Like any skill, polynomial division becomes easier with practice. Let's look at some additional examples and exercises to solidify your understanding.

Example 1

Divide ${x^3 - 8}$ by ${x - 2}$.

Solution:

Notice that we're missing ${x^2}$ and ${x}$ terms in the dividend. We'll need to add placeholders:

${\frac{x^3 + 0x^2 + 0x - 8}{x - 2}}$

Follow the steps outlined above, and you'll find the quotient is ${x^2 + 2x + 4}$ with no remainder. So, the answer is:

${x^2 + 2x + 4}$

Example 2

Divide ${3x^4 - 2x^3 + 5x - 1}$ by ${x + 1}$.

Solution:

Again, we have a missing ${x^2}$ term. Include the placeholder:

${\frac{3x^4 - 2x^3 + 0x^2 + 5x - 1}{x + 1}}$

After performing the division, you should get a quotient of ${3x^3 - 5x^2 + 5x}$ and a remainder of ${-6}$. The answer in the correct format is:

${3x^3 - 5x^2 + 5x + \frac{-6}{x + 1}}$

Exercises

Try these exercises on your own to practice:

1. ${\frac{x^3 + 2x^2 - 5x - 6}{x - 2}}$
2. ${\frac{2x^4 - x^3 + 3x - 1}{x + 1}}$
3. ${\frac{x^3 - 27}{x - 3}}$

Check your answers by multiplying the quotient by the divisor and adding the remainder. You should get the original dividend.

Conclusion: Mastering Polynomial Division

Polynomial division might seem tricky at first, but with a systematic approach and plenty of practice, you can master this important algebraic skill. Remember to set up the problem correctly, follow the steps carefully, pay attention to signs, and include placeholders when necessary. By working through examples and exercises, you'll build confidence and develop a solid understanding of polynomial division. Keep practicing, and you'll be dividing polynomials like a pro in no time! Remember guys, practice is the key to success. Good luck!