Expanding Polynomials A Step-by-Step Guide To (7x^2)(2x^3+5)(x^2-4x-9)
Calculating the product of polynomials might seem daunting at first, but with a systematic approach, it becomes a manageable task. In this article, we will delve into the process of multiplying polynomials, focusing on the expression (7x2)(2x3+5)(x^2-4x-9). We'll break down each step, providing clear explanations and practical tips to help you master polynomial multiplication.
Expanding the Expression: A Step-by-Step Approach
To begin, let's identify the core challenge: effectively expanding the product (7x2)(2x3+5)(x^2-4x-9). This involves multiplying three polynomial expressions, which can be simplified by tackling it in stages. First, we'll multiply the monomial (7x^2) with the binomial (2x^3+5). Then, we'll take the resulting expression and multiply it with the trinomial (x^2-4x-9). This stepwise approach ensures accuracy and clarity.
Step 1: Multiplying the Monomial by the Binomial
The initial step is to multiply the monomial 7x^2 by the binomial 2x^3 + 5. We achieve this by applying the distributive property, a fundamental concept in algebra. The distributive property states that a(b + c) = ab + ac. In our case, a is 7x^2, b is 2x^3, and c is 5. Applying this property, we get:
7x^2 * (2x^3 + 5) = (7x^2 * 2x^3) + (7x^2 * 5)
Now, we multiply the terms individually. Remember the rule for multiplying exponents: when multiplying terms with the same base, we add the exponents. Thus, x^2 * x^3 = x^(2+3) = x^5. Applying this rule:
(7x^2 * 2x^3) = 14x^5
And:
(7x^2 * 5) = 35x^2
Combining these results, the product of the monomial and the binomial is:
14x^5 + 35x^2
This simplifies our original expression, setting the stage for the next step.
Step 2: Multiplying the Result by the Trinomial
Having multiplied the monomial and binomial, we now face the task of multiplying the resulting expression, 14x^5 + 35x^2, by the trinomial, x^2 - 4x - 9. This step again utilizes the distributive property, but on a larger scale. We must distribute each term of the first expression across all terms of the second expression. This process is crucial for correctly expanding the polynomial product.
We begin by multiplying 14x^5 by each term in the trinomial:
14x^5 * (x^2 - 4x - 9) = (14x^5 * x^2) - (14x^5 * 4x) - (14x^5 * 9)
Applying the exponent rule, we get:
14x^7 - 56x^6 - 126x^5
Next, we multiply 35x^2 by each term in the trinomial:
35x^2 * (x^2 - 4x - 9) = (35x^2 * x^2) - (35x^2 * 4x) - (35x^2 * 9)
Again, applying the exponent rule:
35x^4 - 140x^3 - 315x^2
Step 3: Combining Like Terms for the Final Expression
With each term in the first expression multiplied by each term in the second, we arrive at a lengthy expression. The final step is to combine like terms. Like terms are those with the same variable and exponent. This simplification is essential for presenting the polynomial in its most concise form.
Our expanded expression currently looks like this:
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
Upon inspection, we find that there are no like terms in this expression. Each term has a unique power of x. Therefore, the expression is already in its simplest form. This final result represents the expanded product of the original polynomials.
The Final Result
Therefore, the product of the polynomials (7x2)(2x3+5)(x^2-4x-9) is:
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
This is the fully expanded form of the given expression.
Key Concepts and Techniques Revisited
Let's revisit the key concepts and techniques we've employed to successfully multiply these polynomials. These strategies are fundamental to polynomial manipulation and will serve you well in more complex algebraic problems. We focused on the distributive property and the rules for multiplying exponents, both crucial for accurate expansion.
The Distributive Property: The Foundation of Polynomial Multiplication
The distributive property is the cornerstone of polynomial multiplication. As mentioned earlier, it allows us to multiply a single term by a group of terms within parentheses. By distributing each term across the others, we ensure that the entire expression is accurately expanded. In our example, we applied the distributive property twice: first to multiply the monomial by the binomial, and then to multiply the resulting expression by the trinomial. This systematic application is key to handling larger polynomial products.
Exponent Rules: Simplifying the Process
Exponent rules play a vital role in simplifying polynomial multiplication. Specifically, the rule that states x^m * x^n = x^(m+n) is frequently used. This rule allows us to efficiently combine terms with the same base. Without a solid understanding of exponent rules, polynomial multiplication can become cumbersome and error-prone. In our example, we applied this rule multiple times to combine terms like x^2 * x^3 and x^5 * x^2.
Combining Like Terms: Achieving the Simplest Form
After expanding the expression, combining like terms is the final step in simplification. Like terms, as we've defined, have the same variable and exponent. By combining these terms, we reduce the expression to its most concise form. This not only makes the expression easier to read but also simplifies further calculations or manipulations that might be required. In our specific example, we were fortunate that no like terms existed after expansion, meaning the expression was already in its simplest form.
Common Mistakes and How to Avoid Them
When multiplying polynomials, several common mistakes can occur. Awareness of these pitfalls is the first step in avoiding them. Let's discuss some of these errors and strategies for ensuring accuracy.
Forgetting to Distribute to All Terms
A frequent mistake is failing to distribute a term to every term within the parentheses. This often happens when dealing with longer expressions. To avoid this, always double-check that each term has been multiplied correctly. A systematic approach, such as writing out each multiplication step, can be helpful.
Errors with Exponent Rules
Mistakes with exponent rules are another common source of errors. For instance, students might mistakenly multiply exponents instead of adding them. Regular practice and careful attention to the rules can help prevent these errors. It's also beneficial to write out the exponent rule explicitly during the calculation process as a reminder.
Sign Errors
Sign errors, particularly with negative signs, are also common. A misplaced negative sign can drastically alter the result. Pay close attention to the signs of each term and use parentheses to maintain clarity. When distributing a negative term, be sure to apply the negative sign to every term within the parentheses.
Skipping Steps
Rushing through the multiplication process and skipping steps can lead to errors. It's better to take your time and write out each step clearly. This allows you to track your work and identify any mistakes more easily. Showing your work also makes it easier for others to follow your solution and offer assistance if needed.
Practice Problems
To solidify your understanding of polynomial multiplication, let's consider a few practice problems. Working through these examples will reinforce the concepts and techniques we've discussed.
Practice Problem 1
Multiply: (3x)(x^2 - 2x + 1)
This problem involves multiplying a monomial by a trinomial. Apply the distributive property and exponent rules to expand the expression. Remember to combine like terms at the end.
Practice Problem 2
Multiply: (x + 2)(x - 3)
Here, we multiply two binomials. The distributive property still applies, but you'll need to distribute each term of the first binomial across the second. This is often referred to as the FOIL method (First, Outer, Inner, Last).
Practice Problem 3
Multiply: (2x^2 + 1)(x^2 - 4)
This problem involves multiplying two binomials with higher-degree terms. Pay close attention to the exponent rules and combine like terms carefully.
By working through these practice problems, you'll gain confidence in your ability to multiply polynomials accurately and efficiently.
Conclusion
In conclusion, multiplying polynomials requires a systematic approach, a solid understanding of the distributive property and exponent rules, and careful attention to detail. By following the steps outlined in this article and practicing regularly, you can master polynomial multiplication and confidently tackle more complex algebraic problems. Remember to avoid common mistakes by taking your time, showing your work, and double-checking your calculations. With practice, you'll find that multiplying polynomials becomes a manageable and even enjoyable task.
