Simplifying Polynomial Expressions: A Step-by-Step Guide

by Alex Johnson 57 views

In this article, we'll walk through the process of simplifying a polynomial expression. Specifically, we will address how to find the product of (p3)(2p2−4p)(3p2−1)(p^3)(2p^2-4p)(3p^2-1). Polynomial simplification involves applying the distributive property and combining like terms. Let's dive into a detailed, step-by-step solution.

Understanding Polynomial Expressions

Before we tackle the problem, it's important to understand what polynomial expressions are and how they work. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For instance, 2x2+3x−52x^2 + 3x - 5 is a polynomial. Polynomials are fundamental in algebra and are used extensively in various mathematical and scientific fields.

The key operations we'll be using are:

  • Multiplication: Multiplying terms together.
  • Distribution: Distributing one term across multiple terms within parentheses.
  • Combining Like Terms: Adding or subtracting terms that have the same variable and exponent.

Let's begin by outlining the problem we aim to solve. The given expression is (p3)(2p2−4p)(3p2−1)(p^3)(2p^2-4p)(3p^2-1), and our goal is to simplify it to one of the provided options.

Step-by-Step Solution

To simplify the given expression (p3)(2p2−4p)(3p2−1)(p^3)(2p^2-4p)(3p^2-1), we will follow a systematic approach. This involves first multiplying two of the factors, and then multiplying the result by the remaining factor. Here's how we'll proceed:

Step 1: Multiply the First Two Factors

Let's start by multiplying (p3)(p^3) by (2p2−4p)(2p^2 - 4p). We distribute p3p^3 across both terms inside the parenthesis:

p3∗(2p2−4p)=(p3∗2p2)−(p3∗4p)p^3 * (2p^2 - 4p) = (p^3 * 2p^2) - (p^3 * 4p)

When multiplying terms with exponents, we add the exponents:

2p3+2−4p3+1=2p5−4p42p^{3+2} - 4p^{3+1} = 2p^5 - 4p^4

So, after multiplying the first two factors, we have 2p5−4p42p^5 - 4p^4.

Step 2: Multiply the Result by the Third Factor

Next, we take the result from Step 1, which is (2p5−4p4)(2p^5 - 4p^4), and multiply it by the third factor (3p2−1)(3p^2 - 1). We distribute each term in the first expression across both terms in the second expression:

(2p5−4p4)∗(3p2−1)=(2p5∗3p2)+(2p5∗−1)+(−4p4∗3p2)+(−4p4∗−1)(2p^5 - 4p^4) * (3p^2 - 1) = (2p^5 * 3p^2) + (2p^5 * -1) + (-4p^4 * 3p^2) + (-4p^4 * -1)

Now, let's perform each multiplication:

  • 2p5∗3p2=6p5+2=6p72p^5 * 3p^2 = 6p^{5+2} = 6p^7
  • 2p5∗−1=−2p52p^5 * -1 = -2p^5
  • −4p4∗3p2=−12p4+2=−12p6-4p^4 * 3p^2 = -12p^{4+2} = -12p^6
  • −4p4∗−1=4p4-4p^4 * -1 = 4p^4

Combining these results, we get:

6p7−2p5−12p6+4p46p^7 - 2p^5 - 12p^6 + 4p^4

Step 3: Rearrange the Terms

To present the polynomial in standard form, we rearrange the terms in descending order of their exponents:

6p7−12p6−2p5+4p46p^7 - 12p^6 - 2p^5 + 4p^4

Thus, the simplified expression is 6p7−12p6−2p5+4p46p^7 - 12p^6 - 2p^5 + 4p^4.

Analyzing the Options

Now that we have simplified the expression, let's compare our result with the given options:

A. 6p7+4p46p^7 + 4p^4 B. 6p7−2p5−12p5+4p36p^7 - 2p^5 - 12p^5 + 4p^3 C. 6p7−12p6−2p5+4p46p^7 - 12p^6 - 2p^5 + 4p^4 D. 6p12−14p6+4p36p^{12} - 14p^6 + 4p^3

Comparing our simplified expression 6p7−12p6−2p5+4p46p^7 - 12p^6 - 2p^5 + 4p^4 with the options, we can see that option C matches perfectly.

Therefore, the correct answer is:

C. 6p7−12p6−2p5+4p46p^7 - 12p^6 - 2p^5 + 4p^4

Common Mistakes to Avoid

When simplifying polynomial expressions, there are several common mistakes to watch out for. Avoiding these mistakes can significantly improve accuracy.

  • Incorrectly Applying the Distributive Property: Ensure that each term inside the parentheses is multiplied by the term outside.
  • Mistakes with Exponents: When multiplying terms with the same base, remember to add the exponents, not multiply them.
  • Forgetting to Distribute Negative Signs: Pay close attention to negative signs, especially when distributing across multiple terms.
  • Combining Unlike Terms: Only combine terms that have the same variable and exponent. For example, 2p52p^5 and 4p44p^4 cannot be combined.

By being mindful of these common pitfalls, you can increase your accuracy and confidence in simplifying polynomial expressions.

Practice Problems

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

  1. Simplify: (x2)(3x−2)(x2+1)(x^2)(3x - 2)(x^2 + 1)
  2. Simplify: (2y3)(−y2+5y)(y−3)(2y^3)(-y^2 + 5y)(y - 3)
  3. Simplify: (a4)(2a2−a)(4a2+2)(a^4)(2a^2 - a)(4a^2 + 2)

Working through these problems will give you hands-on experience and help solidify your skills in simplifying polynomial expressions.

Conclusion

In summary, simplifying polynomial expressions involves careful application of the distributive property, correct handling of exponents, and combining like terms. By following a step-by-step approach and being aware of common mistakes, you can confidently simplify complex expressions. The product of (p3)(2p2−4p)(3p2−1)(p^3)(2p^2-4p)(3p^2-1) is 6p7−12p6−2p5+4p46p^7 - 12p^6 - 2p^5 + 4p^4, which corresponds to option C.

For further learning and more complex examples, you might find resources on websites like Khan Academy Algebra incredibly helpful.