Solving Polynomials: A Step-by-Step Guide to Equation Factoring and Quadratic Formula Application
Polynomials are fundamental in mathematics, and understanding how to solve them efficiently can be highly beneficial, especially in fields like engineering, physics, and computer science. In this article, we will delve into the process of solving the polynomial equation z^4 - 6z^3 - 8z^2 0. We will cover the step-by-step process, from factoring out common terms to applying the quadratic formula, and provide clear explanations and examples.
Step 1: Factoring Out Common Terms
Begin by identifying and factoring out the greatest common factor (GCF) from the equation. In this case, the GCF for the terms z^4, -6z^3, and -8z^2 is z^2. Factoring out z^2 from the equation, we get:
z^2 (z^2 - 6z - 8) 0
Step 2: Setting Each Factor to Zero
Next, we set each factor of the equation equal to zero and solve for z.
From z^2 0:
z 0
From z^2 - 6z - 8 0:
Step 3: Using the Quadratic Formula
For the quadratic equation z^2 - 6z - 8 0, we can use the quadratic formula: z (-b ± sqrt(b^2 - 4ac)) / (2a). Here, a 1, b -6, and c -8. Let's calculate the discriminant first:
Discriminant: b^2 - 4ac (-6)^2 - 4(1)(-8) 36 32 68
Now, applying the quadratic formula:
z (-(-6) ± sqrt(68)) / (2 * 1) (6 ± sqrt(68)) / 2
Since sqrt(68) 2*sqrt(17), we have:
z (6 ± 2*sqrt(17)) / 2 (6 2*sqrt(17)) / 2 and (6 - 2*sqrt(17)) / 2
However, for simplicity and completeness, let's simplify this further to get:
z 3 sqrt(17) and z 3 - sqrt(17)
Step 4: Combining All Solutions
Combining all the solutions, we get:
z 0, -2, -4, 3 sqrt(17), 3 - sqrt(17)
Thus, the complete set of solutions to the equation z^4 - 6z^3 - 8z^2 0 is z 0, -2, -4, 3 sqrt(17), 3 - sqrt(17).
Conclusion
In this article, we have explored the methods to solve polynomial equations step-by-step. By factoring out common terms and applying the quadratic formula, we can efficiently find the solutions to complex polynomial equations. Whether you are dealing with quadratic equations or higher-degree polynomials, understanding these methods is essential for any mathematician, scientist, or engineer.