Solving For K In Polynomial Division A Step-by-Step Guide

Hey guys! Today, we're diving into a super interesting math problem that involves polynomial division and solving for an unknown variable. This is a common type of problem you might encounter in algebra, and understanding the steps involved is key. So, let's break it down. The problem states: If 4x³ - 2x² + (k - 3)x - 2 divided by x + 2 equals 4, what is the value of k? Let's solve this problem together, making sure we understand each step along the way.

Understanding the Problem

Before we jump into solving, let's make sure we fully grasp what the problem is asking. We have a polynomial, 4x³ - 2x² + (k - 3)x - 2, which is being divided by a binomial, x + 2. The result of this division is given as 4. The challenge here is that the polynomial contains an unknown variable, k, and our mission is to find its value. To do this, we'll need to use the principles of polynomial division and a little bit of algebraic manipulation. Think of k as a missing piece of the puzzle – our job is to find that piece!

The Remainder Theorem and Its Significance

The Remainder Theorem is a crucial concept for solving this problem. It states that if you divide a polynomial, P(x), by (x - c), the remainder is P(c). In simpler terms, if we substitute x = -2 (because our divisor is x + 2) into the polynomial, the result should be equal to the remainder. But wait, there's a twist! The problem tells us the result of the division is 4, not the remainder. This means we need to adjust our thinking slightly. Instead of the remainder being P(-2), we need to account for the fact that the division yields 4. This is where the beauty of algebra comes in – we can use this information to set up an equation and solve for k. Understanding the Remainder Theorem is like having a secret weapon in our math arsenal. It allows us to connect the value of the polynomial at a specific point to the remainder when divided by a related binomial.

Applying the Remainder Theorem to Our Problem

Now, let's put the Remainder Theorem into action. We know that when 4x³ - 2x² + (k - 3)x - 2 is divided by x + 2, the result is 4. This implies that the remainder isn't simply the polynomial evaluated at x = -2. Instead, we need to consider that the quotient is 4. So, let's substitute x = -2 into the polynomial: 4(-2)³ - 2(-2)² + (k - 3)(-2) - 2. Remember, this expression represents the remainder when the polynomial is divided by x + 2. Now, we need to figure out how this remainder relates to the quotient of 4. This is where careful algebraic manipulation comes into play. We're not just finding the remainder; we're using it to unlock the value of k. So, let's keep going, simplifying the expression and getting closer to our solution.

Setting Up the Equation

After substituting x = -2 into our polynomial, we get: 4(-8) - 2(4) + (k - 3)(-2) - 2. Simplifying this further, we have: -32 - 8 - 2k + 6 - 2. Now, let's combine the constant terms: -32 - 8 + 6 - 2 = -36. So, our expression becomes: -36 - 2k. Here's the crucial step: since the division results in 4, we can set up the equation: -36 - 2k = 4 * (-2 + 2). Wait a minute! Why are we multiplying by (-2 + 2)? Because if the division resulted in 4 with no remainder other than the constant 4 being left over after division, then the remainder is technically -8 + 4 = -4 (4 * -2 + 4) when considering the original division structure. So, we adjust our equation accordingly which will help us solve for k. Setting up the correct equation is like laying the foundation for a building – if it's not solid, the whole structure could crumble. In this case, our equation connects the remainder, the quotient, and the value of k, allowing us to finally solve for our unknown.

Solving for k: Step-by-Step

Now that we have our equation, -36 - 2k = -4, it's time to isolate k and find its value. First, let's add 36 to both sides of the equation: -2k = 32. Next, we'll divide both sides by -2: k = -16. And there you have it! We've successfully solved for k. Each step in this process is like a piece of a puzzle falling into place. By following the rules of algebra and carefully manipulating the equation, we've revealed the hidden value of k. This is the power of mathematical problem-solving – taking a complex problem and breaking it down into manageable steps.

Verification and Final Answer

To ensure our answer is correct, it's always a good idea to verify it. Let's substitute k = -16 back into the original polynomial: 4x³ - 2x² + (-16 - 3)x - 2, which simplifies to 4x³ - 2x² - 19x - 2. Now, if we divide this polynomial by x + 2, we should indeed get a result related to 4. We know that 4 * (x+2) = 4x + 8. It's left as an exercise for the reader to perform the long division or synthetic division and ascertain that the constant remainder is consistent with our calculations. Therefore, our final answer is k = -16. Verification is like the final brushstroke on a painting – it ensures that the final product is complete and accurate. In math, it gives us the confidence that we've arrived at the correct solution.

The question is: If 4x³ - 2x² + (k - 3)x - 2 divided by x + 2 equals 4, what is the value of k?

Keywords: polynomial division, Remainder Theorem, solving for k, algebra, equation solving