Simplifying Polynomial Expressions A Step By Step Guide

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In the realm of mathematics, particularly algebra, simplifying expressions is a fundamental skill. This article will guide you through the process of simplifying a polynomial expression, specifically focusing on the expression (x-4)(x^2+3x-5). We will break down the steps, explain the underlying principles, and arrive at the correct answer. Whether you're a student grappling with algebra or simply looking to refresh your math skills, this guide will provide a clear and concise explanation.

Understanding Polynomial Expressions

Before diving into the simplification process, it's crucial to understand what polynomial expressions are. A polynomial is an expression consisting of variables (also known as indeterminates) and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Examples of polynomials include x^2 + 3x - 5 and x - 4. These expressions are the building blocks of many algebraic equations and are essential in various mathematical and scientific fields.

The expression we're tackling, (x-4)(x^2+3x-5), involves the product of two polynomials. The first polynomial, (x-4), is a binomial, a polynomial with two terms. The second polynomial, (x^2+3x-5), is a trinomial, a polynomial with three terms. To simplify this expression, we need to apply the distributive property, a core concept in algebra.

Applying the Distributive Property

The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. In simpler terms, it means we multiply each term inside the parentheses by the term outside the parentheses. When dealing with polynomials, we extend this property to multiply each term in the first polynomial by each term in the second polynomial. This process is often referred to as the FOIL method (First, Outer, Inner, Last) when multiplying two binomials, but the underlying principle remains the same for polynomials with more terms.

To simplify (x-4)(x^2+3x-5), we'll multiply each term in (x-4) by each term in (x^2+3x-5). Let's break it down step-by-step:

  1. Multiply x from (x-4) by each term in (x^2+3x-5):

    • x * x^2 = x^3
    • x * 3x = 3x^2
    • x * -5 = -5x

  2. Multiply -4 from (x-4) by each term in (x^2+3x-5):

    • -4 * x^2 = -4x^2
    • -4 * 3x = -12x
    • -4 * -5 = 20

Now, we combine the results:

x^3 + 3x^2 - 5x - 4x^2 - 12x + 20

Combining Like Terms

The next crucial step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have x^3, x^2 terms, x terms, and a constant term. We'll group the like terms together:

  • x^3 (only one term)
  • 3x^2 and -4x^2 (x^2 terms)
  • -5x and -12x (x terms)
  • 20 (constant term)

Now, we combine the coefficients of the like terms:

  • 3x^2 - 4x^2 = -x^2
  • -5x - 12x = -17x

So, our expression becomes:

x^3 - x^2 - 17x + 20

Identifying the Correct Answer

Now that we've simplified the expression, we can compare it to the given options:

A. x^3 + 7x^2 - 17x + 20 B. x^3 - x^2 - 6x - 20 C. x^3 - x^2 - 17x + 20 D. x^3 - x^2 + 7x - 20

Our simplified expression, x^3 - x^2 - 17x + 20, matches option C. Therefore, the correct answer is C.

Common Mistakes to Avoid

Simplifying polynomial expressions can be tricky, and there are several common mistakes students often make. Being aware of these pitfalls can help you avoid errors and improve your accuracy.

  1. Incorrectly Applying the Distributive Property: A common mistake is forgetting to multiply every term in the second polynomial by each term in the first polynomial. Make sure you distribute each term thoroughly.
  2. Sign Errors: Pay close attention to the signs (positive and negative) when multiplying and combining terms. A simple sign error can lead to an incorrect answer.
  3. Combining Unlike Terms: Only combine terms that have the same variable raised to the same power. For example, you can combine 3x^2 and -4x^2, but you cannot combine 3x^2 and -5x.
  4. Forgetting to Distribute Negative Signs: When multiplying by a negative term, remember to distribute the negative sign to all terms inside the parentheses. For example, -4 * (x^2+3x-5) = -4x^2 - 12x + 20.

Practice Problems

To solidify your understanding, let's work through a few more practice problems:

  1. Simplify (2x + 1)(x - 3)
  2. Simplify (x + 2)(x^2 - x + 1)
  3. Simplify (3x - 2)(2x^2 + 4x - 5)

Solutions:

  1. (2x + 1)(x - 3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3
  2. (x + 2)(x^2 - x + 1) = x^3 - x^2 + x + 2x^2 - 2x + 2 = x^3 + x^2 - x + 2
  3. (3x - 2)(2x^2 + 4x - 5) = 6x^3 + 12x^2 - 15x - 4x^2 - 8x + 10 = 6x^3 + 8x^2 - 23x + 10

Conclusion

Simplifying polynomial expressions is a crucial skill in algebra. By understanding the distributive property and the process of combining like terms, you can confidently tackle these types of problems. Remember to pay attention to signs, avoid common mistakes, and practice regularly to improve your proficiency. Mastering these skills will not only help you in your math courses but also in various real-world applications where algebraic thinking is essential. The correct answer to our initial problem, simplifying (x-4)(x^2+3x-5), is indeed C. x^3 - x^2 - 17x + 20. Keep practicing, and you'll become a polynomial simplification pro! This comprehensive guide should serve as a valuable resource for anyone looking to understand and simplify polynomial expressions effectively. Always double-check your work and take your time to ensure accuracy. With consistent effort and a solid understanding of the principles, you'll be well-equipped to handle any polynomial simplification challenge that comes your way.