Dividing Polynomials (2x^4 − 2x^2 − 7) By (x^2 + X − 6) A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of polynomial division. Polynomial division might seem daunting at first, but with a step-by-step approach, it becomes a manageable and even enjoyable process. In this guide, we're going to tackle a specific problem: dividing the polynomial 2x^4 − 2x^2 − 7 by x^2 + x − 6. So, grab your pencils and let’s get started!

Understanding Polynomial Division

Before we jump into the problem, let's make sure we're all on the same page about what polynomial division actually is. Think of it like regular long division, but instead of numbers, we're working with expressions containing variables and exponents. The goal is the same: to break down a complex expression into simpler parts.

Polynomial division is a method used to divide a polynomial by another polynomial of lower or equal degree. It's a crucial tool in algebra and calculus, helping us simplify expressions, solve equations, and understand the behavior of polynomial functions. Just like with numerical long division, we have a dividend (the polynomial being divided), a divisor (the polynomial we're dividing by), a quotient (the result of the division), and sometimes a remainder (what's left over if the division isn't perfect).

Why is this important? Well, polynomial division is used in various mathematical applications, including finding roots of polynomials, simplifying rational expressions, and performing integration in calculus. Understanding how to divide polynomials opens doors to solving more complex problems and gaining deeper insights into mathematical concepts. Moreover, mastering polynomial division provides a solid foundation for understanding more advanced algebraic concepts, such as synthetic division and the Remainder Theorem. By grasping the fundamentals of polynomial division, you'll be well-equipped to tackle a wide range of mathematical challenges. So, let's dive in and conquer this essential skill together!

Setting Up the Problem

Okay, let's get to our specific problem: dividing 2x^4 − 2x^2 − 7 by x^2 + x − 6. The first thing we need to do is set up the problem in a way that resembles long division. It's crucial to make sure we include placeholders for any missing terms in the dividend. What do I mean by that? Well, notice that in our dividend, 2x^4 − 2x^2 − 7, we're missing the x^3 and x terms. We need to include them with coefficients of 0 to keep everything aligned properly. So, we'll rewrite the dividend as 2x^4 + 0x^3 − 2x^2 + 0x − 7.

This step is super important because it helps us keep our columns aligned during the division process. Imagine trying to do long division with numbers and forgetting to include a zero in the tens place – you'd quickly get confused! It’s the same idea here. By including these placeholders, we ensure that we're dividing like terms by like terms, which is essential for getting the correct answer. Think of it as creating a roadmap for our division journey. These placeholders act as signposts, guiding us through each step and preventing us from veering off course. Trust me, guys, taking the extra moment to add these zeros will save you headaches down the road.

Now, let's set up the long division. We'll write the divisor, x^2 + x − 6, on the left side and the dividend, 2x^4 + 0x^3 − 2x^2 + 0x − 7, under the division symbol. This setup mirrors the familiar format of numerical long division, but instead of digits, we're working with polynomial terms. This visual representation will help us organize our steps and keep track of the quotient and remainder. Remember, a clear setup is half the battle! Once we have everything in place, we can start the actual division process. So, let's move on to the next step and begin the exciting journey of dividing these polynomials!

Step-by-Step Division Process

Alright, now for the fun part – the actual division! Here’s how we'll tackle dividing 2x^4 + 0x^3 − 2x^2 + 0x − 7 by x^2 + x − 6, step by step:

Step 1: Divide the Leading Terms

Focus on the leading terms of both the dividend (2x^4) and the divisor (x^2). We need to figure out what we need to multiply x^2 by to get 2x^4. The answer is 2x^2. This 2x^2 is the first term of our quotient.

Step 2: Multiply and Subtract

Next, we multiply the entire divisor (x^2 + x − 6) by the 2x^2 we just found. This gives us 2x^4 + 2x^3 − 12x^2. Now, we subtract this result from the corresponding terms of the dividend. Be careful with your signs here – subtraction can be tricky!

 2x^4 + 0x^3 − 2x^2 + 0x − 7
−(2x^4 + 2x^3 − 12x^2)

Subtracting gives us -2x^3 + 10x^2.

Step 3: Bring Down the Next Term

Just like in regular long division, we bring down the next term from the dividend, which is +0x. Our new expression is -2x^3 + 10x^2 + 0x.

Step 4: Repeat the Process

Now we repeat the process. We focus on the leading term of our new expression, -2x^3, and the leading term of the divisor, x^2. What do we multiply x^2 by to get -2x^3? The answer is -2x. This is the next term in our quotient.

Multiply the divisor (x^2 + x − 6) by -2x, which gives us -2x^3 − 2x^2 + 12x. Subtract this from our current expression:

-2x^3 + 10x^2 + 0x
−(-2x^3 − 2x^2 + 12x)

This leaves us with 12x^2 − 12x.

Step 5: Bring Down the Last Term

Bring down the last term from the dividend, which is -7. Our new expression is 12x^2 − 12x − 7.

Step 6: Final Division

Repeat the process one more time. What do we multiply x^2 by to get 12x^2? The answer is 12. This is the final term in our quotient.

Multiply the divisor (x^2 + x − 6) by 12, which gives us 12x^2 + 12x − 72. Subtract this from our current expression:

12x^2 − 12x − 7
−(12x^2 + 12x − 72)

This leaves us with a remainder of -24x + 65.

The Solution: Quotient and Remainder

So, what's our answer? After going through all the steps, we've found that when we divide 2x^4 − 2x^2 − 7 by x^2 + x − 6, we get a quotient of 2x^2 − 2x + 12 and a remainder of -24x + 65. We can write this as:

(2x^4 − 2x^2 − 7) / (x^2 + x − 6) = 2x^2 − 2x + 12 + (-24x + 65) / (x^2 + x − 6)

This is the final solution to our polynomial division problem. Remember, the quotient is the result of the division, and the remainder is what's left over. It's like saying,