Polynomial Division: Solve (2x^3 - 3x^2 - 5x - 12) / (x - 3)
Let's dive into how to divide the polynomial expression (2x^3 - 3x^2 - 5x - 12) by (x - 3). Polynomial division might seem tricky at first, but with a step-by-step approach, it becomes quite manageable. We'll explore the process, break it down, and find the solution together. So, if you're ready to conquer polynomial division, let's get started!
Understanding Polynomial Division
Before we jump into the specific problem, let's quickly recap what polynomial division is all about. Polynomial division is essentially the same as long division you learned back in elementary school, but now we're dealing with expressions containing variables and exponents. It's a method used to divide a polynomial by another polynomial of lower or equal degree. The key is to follow a systematic approach to ensure accuracy. There are two primary methods for polynomial division: long division and synthetic division. For this particular problem, we'll focus on long division as it's generally more versatile and easier to understand conceptually.
To master polynomial division, understanding the terminology is crucial. The polynomial being divided (in our case, 2x^3 - 3x^2 - 5x - 12) is called the dividend. The polynomial we are dividing by (x - 3) is the divisor. The result of the division is the quotient, and any remaining part is the remainder. Our goal is to find the quotient when (2x^3 - 3x^2 - 5x - 12) is divided by (x - 3).
Now, let's talk about why polynomial division is so important. It's not just an abstract mathematical concept; it has numerous applications in various fields. For instance, in calculus, polynomial division helps simplify complex rational functions, making them easier to integrate or differentiate. In engineering, it's used in control systems analysis and design. Furthermore, understanding polynomial division is foundational for more advanced algebraic concepts, like factoring polynomials and solving polynomial equations. So, mastering this skill opens doors to a deeper understanding of mathematics and its applications.
Step-by-Step Solution Using Long Division
Now, let's tackle the problem: divide (2x^3 - 3x^2 - 5x - 12) by (x - 3). We'll use the long division method, breaking it down into easy-to-follow steps.
- Set up the long division: Write the dividend (2x^3 - 3x^2 - 5x - 12) inside the division symbol and the divisor (x - 3) outside. Make sure the terms are arranged in descending order of their exponents.
________________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
- Divide the first term: Divide the first term of the dividend (2x^3) by the first term of the divisor (x). This gives you 2x^2. Write this above the division symbol, aligning it with the x^2 term.
2x^2____________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
- Multiply: Multiply the quotient term (2x^2) by the entire divisor (x - 3). This gives you 2x^3 - 6x^2.
2x^2____________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
2x^3 - 6x^2
- Subtract: Subtract the result (2x^3 - 6x^2) from the corresponding terms of the dividend. This gives you (2x^3 - 3x^2) - (2x^3 - 6x^2) = 3x^2.
2x^2____________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
2x^3 - 6x^2
----------
3x^2
- Bring down the next term: Bring down the next term from the dividend (-5x) and write it next to the result (3x^2).
2x^2____________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
2x^3 - 6x^2
----------
3x^2 - 5x
- Repeat: Repeat the process. Divide the first term of the new result (3x^2) by the first term of the divisor (x). This gives you 3x. Write this above the division symbol, aligning it with the x term.
2x^2 + 3x________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
2x^3 - 6x^2
----------
3x^2 - 5x
- Multiply: Multiply the new quotient term (3x) by the entire divisor (x - 3). This gives you 3x^2 - 9x.
2x^2 + 3x________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
2x^3 - 6x^2
----------
3x^2 - 5x
3x^2 - 9x
- Subtract: Subtract the result (3x^2 - 9x) from the corresponding terms. This gives you (3x^2 - 5x) - (3x^2 - 9x) = 4x.
2x^2 + 3x________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
2x^3 - 6x^2
----------
3x^2 - 5x
3x^2 - 9x
--------
4x
- Bring down the last term: Bring down the last term from the dividend (-12) and write it next to the result (4x).
2x^2 + 3x________________
x - 3 | 2x^3 - 3x^2 - 5x - 12
2x^3 - 6x^2
----------
3x^2 - 5x
3x^2 - 9x
--------
4x - 12
-
Repeat one last time: Divide the first term of the new result (4x) by the first term of the divisor (x). This gives you 4. Write this above the division symbol.
2x^2 + 3x + 4____________ x - 3 | 2x^3 - 3x^2 - 5x - 12 2x^3 - 6x^2 ---------- 3x^2 - 5x 3x^2 - 9x -------- 4x - 12 -
Multiply: Multiply the new quotient term (4) by the entire divisor (x - 3). This gives you 4x - 12.
2x^2 + 3x + 4____________ x - 3 | 2x^3 - 3x^2 - 5x - 12 2x^3 - 6x^2 ---------- 3x^2 - 5x 3x^2 - 9x -------- 4x - 12 4x - 12 -
Subtract: Subtract the result (4x - 12) from the corresponding terms. This gives you (4x - 12) - (4x - 12) = 0.
2x^2 + 3x + 4____________ x - 3 | 2x^3 - 3x^2 - 5x - 12 2x^3 - 6x^2 ---------- 3x^2 - 5x 3x^2 - 9x -------- 4x - 12 4x - 12 -------- 0
Since the remainder is 0, the division is exact.
Identifying the Correct Answer
From the long division, we found that (2x^3 - 3x^2 - 5x - 12) divided by (x - 3) is 2x^2 + 3x + 4.
Therefore, the correct answer is:
A. 2x^2 + 3x + 4
So, there you have it! We've successfully divided the polynomial expression using long division. It's all about breaking the problem down into manageable steps and staying organized. Keep practicing, and you'll become a polynomial division pro in no time! Remember, math isn't just about getting the right answer; it's about understanding the process and building a strong foundation for future learning.