Dividing Polynomials: Solve (3x^2 + 11x - 4) / (x + 4)
Hey guys! Let's dive into solving this polynomial division problem. We're going to figure out the result of dividing (3x^2 + 11x - 4) by (x + 4). Polynomial division might seem intimidating at first, but with a step-by-step approach, it becomes super manageable. So, let's break it down and find the correct answer together!
Understanding Polynomial Division
Polynomial division is similar to long division with numbers, but instead of digits, we're dealing with terms containing variables and exponents. The goal is to find a quotient and possibly a remainder when dividing one polynomial by another. Before we jump into solving our specific problem, let's quickly recap the general process. Think of it as dividing a big number by a smaller number, but with algebraic expressions. You need to find out how many times the divisor fits into the dividend, and polynomial division helps us do just that with polynomials.
When performing polynomial division, there are a few key terms to keep in mind. The dividend is the polynomial being divided (in our case, 3x^2 + 11x - 4). The divisor is the polynomial we are dividing by (here, x + 4). The quotient is the result of the division, and the remainder is what's left over, if anything. The process involves dividing, multiplying, subtracting, and bringing down terms until we can't divide any further. It's a systematic way to break down complex polynomial expressions into simpler forms, which is incredibly useful in algebra and calculus. Mastering polynomial division opens doors to solving more advanced problems and understanding the relationships between different polynomial expressions. So, let's get started with our specific problem and see how this works in practice!
Solving (3x^2 + 11x - 4) ÷ (x + 4)
Okay, let's tackle this problem head-on! We want to divide 3x^2 + 11x - 4 by x + 4. We can use either long division or synthetic division. For this explanation, let's use long division since it's a bit more intuitive for those who are just getting started. Synthetic division is faster, but long division helps visualize what’s happening at each step. Trust me, once you get the hang of long division, you'll feel like a polynomial division pro!
Step-by-Step Long Division
- Set up the long division:
__________
x + 4 | 3x^2 + 11x - 4
- Divide the first term of the dividend by the first term of the divisor:
3x^2 ÷ x = 3x. This is the first term of our quotient.
3x_________
x + 4 | 3x^2 + 11x - 4
- Multiply the divisor by the first term of the quotient:
3x * (x + 4) = 3x^2 + 12x
3x_________
x + 4 | 3x^2 + 11x - 4
-(3x^2 + 12x)
- Subtract the result from the dividend:
(3x^2 + 11x) - (3x^2 + 12x) = -x
3x_________
x + 4 | 3x^2 + 11x - 4
-(3x^2 + 12x)
-------------
-x - 4
- Bring down the next term from the dividend:
Bring down the -4 to get -x - 4
3x_________
x + 4 | 3x^2 + 11x - 4
-(3x^2 + 12x)
-------------
-x - 4
- Divide the new first term (-x) by the first term of the divisor (x):
-x ÷ x = -1. This is the next term of our quotient.
3x - 1_____
x + 4 | 3x^2 + 11x - 4
-(3x^2 + 12x)
-------------
-x - 4
- Multiply the divisor by the new term of the quotient:
-1 * (x + 4) = -x - 4
3x - 1_____
x + 4 | 3x^2 + 11x - 4
-(3x^2 + 12x)
-------------
-x - 4
-(-x - 4)
- Subtract the result:
(-x - 4) - (-x - 4) = 0
3x - 1_____
x + 4 | 3x^2 + 11x - 4
-(3x^2 + 12x)
-------------
-x - 4
-(-x - 4)
-------------
0
Since the remainder is 0, the division is exact.
The Solution
The quotient is 3x - 1. Therefore, $(3 x^2+11 x-4) ÷(x+4) = 3x - 1$.
So, the correct answer is:
A. $3x - 1$
Why This Matters
You might be wondering, why bother with polynomial division? Well, it's a fundamental skill in algebra and calculus. It helps in simplifying complex expressions, solving equations, and understanding the behavior of polynomial functions. For example, in calculus, polynomial division can be used to find limits and integrals more easily. In engineering and physics, these skills can be applied to model and solve real-world problems involving polynomial relationships. Polynomial division is also crucial in computer science, particularly in areas like cryptography and coding theory, where polynomials are used to encode and decode information.
Moreover, understanding polynomial division helps develop critical thinking and problem-solving skills. It teaches you to break down complex problems into smaller, manageable steps, and to apply logical reasoning to find solutions. These skills are valuable not only in mathematics but also in many other areas of life. The ability to analyze and solve problems systematically is a key attribute for success in any field. So, mastering polynomial division is not just about getting the right answer; it's about developing a valuable skill set that will serve you well in the long run. Keep practicing, and you'll find that it becomes second nature!
Additional Tips for Polynomial Division
To really nail polynomial division, here are a few extra tips to keep in mind:
- Always check if the dividend and divisor are in descending order of powers. If not, rearrange them before starting the division.
- If there's a missing term (e.g., no x term), include it with a coefficient of 0. This helps keep your columns aligned and prevents mistakes.
- Double-check your multiplication and subtraction steps. These are common areas for errors, so take your time and be careful.
- Practice, practice, practice! The more you do it, the more comfortable you'll become with the process. Try different problems with varying degrees of complexity.
And remember, even if you make a mistake, don't get discouraged! Everyone makes mistakes when learning something new. The key is to learn from those mistakes and keep moving forward. With a little patience and persistence, you'll be dividing polynomials like a pro in no time!