Dividing Polynomials: Easy Step-by-Step Guide

Let's break down how to divide the polynomial $28x^3 + 42x^2 - 35x$ by the monomial $7x$. This is a common type of problem in algebra, and understanding the process is crucial for mastering polynomial manipulation. We'll go through each step meticulously, ensuring clarity and comprehension.

Understanding the Basics

Before we dive into the problem, it's important to understand the basic principles of polynomial division. When we divide a polynomial by a monomial, we're essentially dividing each term of the polynomial by the monomial. This utilizes the distributive property in reverse. Remember that when dividing terms with exponents, you subtract the exponents if the bases are the same (e.g., $x^a / x^b = x^{a-b}$). Also, keep in mind the rules of signs: a positive divided by a positive is positive, a negative divided by a positive is negative, and so on.

Key Concepts to Remember:

• Polynomial: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Monomial: A polynomial with only one term.
• Distributive Property: $a(b + c) = ab + ac$
• Exponent Rules: $x^a / x^b = x^{a-b}$

With these concepts in mind, we can approach the division problem systematically and accurately. Let's get started!

Step-by-Step Division

Now, let's divide $28x^3 + 42x^2 - 35x$ by $7x$. We'll divide each term of the polynomial by $7x$ individually.

1. Divide the first term:

   • Divide $28x^3$ by $7x$.
   • $28x^3 / 7x = (28/7) * (x^3/x) = 4x^{3-1} = 4x^2$

2. Divide the second term:

   • Divide $42x^2$ by $7x$.
   • $42x^2 / 7x = (42/7) * (x^2/x) = 6x^{2-1} = 6x$

3. Divide the third term:

   • Divide $-35x$ by $7x$.
   • $-35x / 7x = (-35/7) * (x/x) = -5x^{1-1} = -5 * 1 = -5$

4. Combine the results:

   • Now, add the results from each division:
   • $4x^2 + 6x - 5$

Therefore, $(28x^3 + 42x^2 - 35x) / (7x) = 4x^2 + 6x - 5$.

Detailed Explanation of Each Step:

• Dividing $28x^3$ by $7x$: When dividing $28x^3$ by $7x$, we first divide the coefficients: $28 / 7 = 4$. Then, we divide the variables: $x^3 / x = x^{3-1} = x^2$. Combining these, we get $4x^2$. This step is all about simplifying the first term of the polynomial by applying the division rules of both coefficients and exponents.

• Dividing $42x^2$ by $7x$: Similarly, for the second term, we divide $42x^2$ by $7x$. The coefficients divide as $42 / 7 = 6$, and the variables divide as $x^2 / x = x^{2-1} = x$. Thus, we have $6x$. This step mirrors the first, ensuring each term is correctly simplified through division.

• Dividing $-35x$ by $7x$: Finally, we divide $-35x$ by $7x$. Here, the coefficients divide as $-35 / 7 = -5$, and the variables divide as $x / x = 1$. Therefore, the result is $-5$. This completes the division process for all terms in the polynomial.

Identifying the Correct Answer

Comparing our result, $4x^2 + 6x - 5$, with the given options:

A. $4x^2 - 6x + 5$ B. $4x^2 + 6x - 5$ C. $4x^3 - 6x^2 + 5$ D. $4x^3 + 6x^2 - 5$

We can see that option B, $4x^2 + 6x - 5$, matches our solution perfectly.

Therefore, the correct answer is B.

Common Mistakes to Avoid

When dividing polynomials, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Forgetting to divide every term: Make sure you divide each term in the polynomial by the monomial. It's a common mistake to only divide the first term and then forget about the others.
• Incorrectly applying exponent rules: Remember that when dividing terms with exponents, you subtract the exponents. For example, $x^5 / x^2 = x^3$, not $x^{5/2}$.
• Sign errors: Pay close attention to the signs of the terms. A negative divided by a positive is negative, and so on.
• Not simplifying completely: Always simplify your answer as much as possible. For example, if you have a term like $6x/2$, simplify it to $3x$.
• Rushing through the process: Take your time and double-check your work. Polynomial division can be tricky, so it's important to be methodical.

By being aware of these common mistakes, you can increase your chances of getting the correct answer.

Practice Problems

To solidify your understanding, here are a few practice problems:

1. Divide $15x^4 + 25x^3 - 30x^2$ by $5x$.
2. Divide $12x^5 - 18x^3 + 24x$ by $6x$.
3. Divide $21x^6 + 14x^4 - 49x^2$ by $7x$.

Solutions:

1. $3x^3 + 5x^2 - 6x$
2. $2x^4 - 3x^2 + 4$
3. $3x^5 + 2x^3 - 7x$

Practice these problems, and you'll become more confident in your ability to divide polynomials.

Conclusion

Dividing polynomials by monomials is a fundamental skill in algebra. By understanding the basic principles, following the steps carefully, and avoiding common mistakes, you can master this concept. Remember to divide each term individually, apply the exponent rules correctly, and pay attention to signs. With practice, you'll become proficient at polynomial division, which will be invaluable in more advanced math courses.

So there you have it, guys! You've now got a solid handle on dividing polynomials by monomials. Keep practicing, and you'll be a pro in no time. Good luck, and happy dividing!

Remember, mathematics is not just about getting the right answer; it's about understanding the process. Each problem is a step towards building a stronger foundation in algebra.

Keep learning and keep growing!