Simplifying Algebraic Expressions: A Step-by-Step Guide

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Hey guys! Let's dive into simplifying algebraic expressions. We'll tackle a specific problem and break down the process so you can understand it inside and out. The ability to simplify expressions is super important in algebra, helping you solve equations and understand mathematical relationships more easily. We'll use a combination of techniques, including polynomial division or synthetic division, to make the expression simpler. So, buckle up, and let's get started.

The Problem: Simplifying a Rational Expression

Okay, so here’s the problem we're going to simplify: $ rac{6 z^3+35 z^2-18 z+2}{z- rac{1}{6}}$. Looks a little intimidating, right? Don't sweat it; we'll take it one step at a time. This is a rational expression, meaning it's a fraction with polynomials in the numerator and denominator. Our goal is to find an equivalent expression that's simpler. In this case, we can use polynomial division. When you're dealing with rational expressions like this, the most common strategy is to try to cancel out any common factors between the numerator and the denominator. Since the denominator is a linear expression (z - 1/6), we can try to divide the numerator by it. This is where polynomial division or synthetic division come in handy. The idea is to rewrite the numerator as the product of the denominator and another polynomial (the quotient) plus a remainder. If the remainder is zero, then the denominator is a factor of the numerator, and we can simplify the fraction. The good news is this is exactly what we're going to be doing. Let's get to work, shall we? To do this correctly, we'll use long division. In the long division method, we'll put the numerator inside the division symbol and the denominator outside. After we perform the long division, we should have an easier expression to work with. We need to pay close attention to the signs and coefficients during the division process. Remember, a small mistake here can lead to a completely wrong answer. We'll make sure our final answer is in simplest form, which means all the terms are combined, and there are no more common factors in the numerator and the denominator. The ability to simplify expressions is not just a skill for getting answers; it's a fundamental part of understanding math.

Step-by-Step Solution: Long Division

Alright, let's get our hands dirty and do some long division. We'll divide the polynomial in the numerator (6z3+35z2−18z+26z^3 + 35z^2 - 18z + 2) by the linear expression in the denominator (z - rac{1}{6}). Long division can look a bit cumbersome at first, but it's a systematic process that helps us break down the problem.

  1. Set up the long division: Write the division problem as follows:
       ________
z - 1/6 | 6z^3 + 35z^2 - 18z + 2
  1. Divide the first term: Divide the first term of the numerator (6z36z^3) by the first term of the denominator (zz). This gives us 6z26z^2. Write this above the division symbol.
       6z^2______
z - 1/6 | 6z^3 + 35z^2 - 18z + 2
  1. Multiply and subtract: Multiply 6z26z^2 by the entire denominator (z - rac{1}{6}), which yields 6z3−z26z^3 - z^2. Write this result below the numerator and subtract:
       6z^2______
z - 1/6 | 6z^3 + 35z^2 - 18z + 2
       -(6z^3 - z^2)
       __________
              36z^2 - 18z
  1. Bring down the next term: Bring down the next term from the numerator (-18z).
       6z^2______
z - 1/6 | 6z^3 + 35z^2 - 18z + 2
       -(6z^3 - z^2)
       __________
              36z^2 - 18z
  1. Repeat the process: Divide the first term of the new expression (36z236z^2) by zz, which gives us 36z36z. Write this above the division symbol.
       6z^2 + 36z_____
z - 1/6 | 6z^3 + 35z^2 - 18z + 2
       -(6z^3 - z^2)
       __________
              36z^2 - 18z
  1. Multiply and subtract again: Multiply 36z36z by the denominator (z - rac{1}{6}), which gives us 36z2−6z36z^2 - 6z. Write this below 36z2−18z36z^2 - 18z and subtract:
       6z^2 + 36z_____
z - 1/6 | 6z^3 + 35z^2 - 18z + 2
       -(6z^3 - z^2)
       __________
              36z^2 - 18z
              -(36z^2 - 6z)
              __________
                      -12z + 2
  1. Bring down the last term: Bring down the last term from the numerator (+2).
       6z^2 + 36z_____
z - 1/6 | 6z^3 + 35z^2 - 18z + 2
       -(6z^3 - z^2)
       __________
              36z^2 - 18z
              -(36z^2 - 6z)
              __________
                      -12z + 2
  1. Final division: Divide −12z-12z by zz, which gives us −12-12. Write this above the division symbol.
       6z^2 + 36z - 12
z - 1/6 | 6z^3 + 35z^2 - 18z + 2
       -(6z^3 - z^2)
       __________
              36z^2 - 18z
              -(36z^2 - 6z)
              __________
                      -12z + 2
  1. Multiply and subtract one last time: Multiply −12-12 by the denominator (z - rac{1}{6}), which gives us −12z+2-12z + 2. Write this below −12z+2-12z + 2 and subtract:

           6z^2 + 36z - 12
    

z - 1/6 | 6z^3 + 35z^2 - 18z + 2 -(6z^3 - z^2) __________ 36z^2 - 18z -(36z^2 - 6z) __________ -12z + 2 -(-12z + 2) _________ 0


We get a remainder of 0.



## The Simplified Expression

Since the remainder is zero, it means that $(z - rac{1}{6})$ is a factor of the numerator. When we divide $6z^3 + 35z^2 - 18z + 2$ by $z - rac{1}{6}$, we get a quotient of $6z^2 + 36z - 12$. Therefore, the simplified expression is:

$rac{6 z^3+35 z^2-18 z+2}{z-rac{1}{6}} = 6z^2 + 36z - 12$

Wow, look at that!  The original expression, which seemed complicated, has been simplified to a much cleaner and more manageable form. This simplified form is equivalent to the original expression, but it's much easier to work with. This means you can plug in values for 'z' into either the original or the simplified version, and you'll get the same result (as long as z is not equal to 1/6). The power of simplification lies in its ability to reveal underlying patterns, make calculations easier, and provide a clearer understanding of the mathematical relationships involved. Always remember, simplifying an expression doesn't change its value; it just makes it look different. Therefore, our final answer is: $6z^2 + 36z - 12$. We have successfully simplified the given expression.  Great job, everyone! 



## Key Takeaways and Next Steps

**Key Takeaways:**

*   **Polynomial Division:** Polynomial division (or synthetic division) is a powerful tool for simplifying rational expressions. It helps us identify factors and simplify expressions that might seem complex at first glance.
*   **Remainders:** A remainder of zero indicates that the denominator is a factor of the numerator, allowing for complete simplification.
*   **Simplification:** Simplifying expressions makes them easier to work with, solve, and understand.

**Next Steps:**

1.  **Practice:**  Try simplifying other rational expressions on your own. The more you practice, the better you'll become at recognizing patterns and applying the techniques.
2.  **Synthetic Division:** Explore synthetic division as an alternative method for dividing polynomials. It's a faster method when you're dividing by a linear expression.
3.  **Applications:** Consider how simplifying expressions can be used in solving equations, graphing functions, and other areas of algebra. Understanding how to simplify complex expressions is not only helpful in math class, but also in a wide range of real-world applications. From engineering to economics, the ability to manipulate and simplify expressions is an invaluable skill. By mastering the methods shown above, you are building a solid foundation in algebra, opening doors to more advanced mathematical concepts and applications. Keep practicing, keep learning, and you'll continue to grow your mathematical skills. Good luck, and keep up the fantastic work!

I hope this step-by-step guide has been helpful, guys! Keep practicing, and you'll become a pro at simplifying algebraic expressions in no time.