Randy's Polynomial Division Error A Step-by-Step Explanation
Hey everyone! Ever get that feeling when you're doing a math problem and something just doesn't quite click? That's what we're diving into today. We're going to break down a polynomial division problem where our friend Randy made a little slip-up. Math is all about learning from these moments, so let's get started!
The Problem: Polynomial Long Division
Randy was tasked with dividing the polynomial 2x⁴ - 3x³ - 3x² + 7x - 3 by x² - 2x + 1. Polynomial long division can be a bit tricky, kind of like a regular long division but with more letters and exponents involved. Let's look at how Randy set up the problem:
x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3
This looks like a classic long division setup, but with polynomials instead of numbers. Now, let’s dissect Randy's attempt and pinpoint where things went sideways.
Randy's Attempt: Spotting the Mistake
Randy's work looks like this:
2x²
x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3
2x⁴ - 4x³ + 2x²
------------------
x³ - 5x² + 7x - 3
At first glance, it seems like Randy's on the right track. He correctly identified that 2x² multiplied by x² gives 2x⁴, which is what we need to start canceling out terms. The first multiplication step, 2x² * (x² - 2x + 1) = 2x⁴ - 4x³ + 2x², is also spot-on. However, the devil is in the details, guys, so let’s slow it down and really inspect what happened after this.
Now comes the subtraction. Randy subtracted 2x⁴ - 4x³ + 2x² from 2x⁴ - 3x³ - 3x². Remember, subtracting polynomials means changing the signs of the terms being subtracted and then combining like terms. So, it should look like this:
(2x⁴ - 3x³ - 3x²) - (2x⁴ - 4x³ + 2x²) = 2x⁴ - 3x³ - 3x² - 2x⁴ + 4x³ - 2x²
Combining these terms, we get:
(2x⁴ - 2x⁴) + (-3x³ + 4x³) + (-3x² - 2x²) = x³ - 5x²
So far, so good! Randy got this part right. He then brought down the next term, +7x, resulting in the new polynomial x³ - 5x² + 7x. And then brought down -3 resulting in x³ - 5x² + 7x - 3. This is where the critical point of the error occurs, so pay close attention!
Randy needed to figure out what term, when multiplied by x² - 2x + 1, would give him a leading term of x³. The correct term is +x. This is because x * x² = x³. However, let’s see what happens when we continue Randy's (incorrect) method and then we’ll pinpoint where he actually went wrong.
Let's assume, for the sake of argument, that Randy continued correctly (which he didn't, but bear with me!). If he correctly identified +x as the next term in the quotient, he would then multiply x by the entire divisor (x² - 2x + 1):
x * (x² - 2x + 1) = x³ - 2x² + x
This result would then be subtracted from x³ - 5x² + 7x - 3:
(x³ - 5x² + 7x - 3) - (x³ - 2x² + x) = x³ - 5x² + 7x - 3 - x³ + 2x² - x
Combining like terms:
(x³ - x³) + (-5x² + 2x²) + (7x - x) - 3 = -3x² + 6x - 3
Now, we need to figure out the next term in the quotient. What term, when multiplied by x², gives us -3x²? The answer is -3. So, we multiply -3 by the divisor:
-3 * (x² - 2x + 1) = -3x² + 6x - 3
And subtract this from -3x² + 6x - 3:
(-3x² + 6x - 3) - (-3x² + 6x - 3) = -3x² + 6x - 3 + 3x² - 6x + 3 = 0
So, if Randy had done everything correctly, the final quotient would be 2x² + x - 3 and the remainder would be 0. This means the division should work out perfectly.
Where Did Randy Go Wrong?
Okay, so after all that, let’s pinpoint the exact moment Randy made his error. Looking back at the work, the mistake wasn't immediately obvious, but it's in the crucial step after bringing down the 7x. Let's revisit that:
Randy had x³ - 5x² + 7x - 3. He needed to divide this by x² - 2x + 1. The question is: what multiplies x² to give x³? It's x, as we discussed. However, in Randy's actual work, he seems to have skipped this step or miscalculated what to do next. This missing step is the key error.
To be precise, after obtaining x³ - 5x² + 7x - 3, Randy should have included +x in the quotient and proceeded with the multiplication and subtraction. By omitting this term, the subsequent steps were based on an incorrect setup, leading to the wrong answer. He didn't continue the long division process correctly by not identifying the next term in the quotient.
Correcting Randy's Mistake: A Step-by-Step Solution
To make sure we’re crystal clear on how this should be done, let’s walk through the corrected polynomial long division step-by-step. This way, we can see exactly how it's meant to flow, and you guys can use this as a reference for similar problems.
-
Set up the division:
x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -
Divide the leading terms:
- 2x⁴ divided by x² is 2x². Write 2x² above the division bar.
2x² x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -
Multiply the divisor by 2x²:
- 2x² * (x² - 2x + 1) = 2x⁴ - 4x³ + 2x²
-
Subtract the result from the dividend:
2x² x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²)- This simplifies to: x³ - 5x² + 7x - 3
-
Bring down the next term (+7x):
2x² x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) ------------------ x³ - 5x² + 7x - 3 -
Divide the new leading term:
- x³ divided by x² is x. Write +x above the division bar.
2x² + x x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) ------------------ x³ - 5x² + 7x - 3 -
Multiply the divisor by x:
- x * (x² - 2x + 1) = x³ - 2x² + x
-
Subtract the result from the current polynomial:
2x² + x x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) ------------------ x³ - 5x² + 7x - 3 -(x³ - 2x² + x)- This simplifies to: -3x² + 6x - 3
-
Bring down the next term (-3):
(We already brought it down in step 5, so this is just a reminder that it's there.)
-
Divide the new leading term:
- -3x² divided by x² is -3. Write -3 above the division bar.
2x² + x - 3 x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) ------------------ x³ - 5x² + 7x - 3 -(x³ - 2x² + x) ------------------ -3x² + 6x - 3 -
Multiply the divisor by -3:
- -3 * (x² - 2x + 1) = -3x² + 6x - 3
-
Subtract the result from the current polynomial:
2x² + x - 3 x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) ------------------ x³ - 5x² + 7x - 3 -(x³ - 2x² + x) ------------------ -3x² + 6x - 3 -(-3x² + 6x - 3)- This simplifies to: 0
-
The Result:
- The quotient is 2x² + x - 3, and the remainder is 0.
So, the correct division yields 2x² + x - 3. This step-by-step breakdown should give a solid understanding of how to tackle these problems.
Key Takeaways: Mastering Polynomial Division
Polynomial division can seem daunting, but like any math skill, it becomes easier with practice and a clear understanding of the steps. Here are some key things to remember to avoid the kind of mistake Randy made:
- Pay close attention to each step: Each term in the quotient needs to be carefully determined by dividing the leading term of the current polynomial by the leading term of the divisor.
- Don’t skip terms: Make sure you bring down all the necessary terms and account for them in your calculations.
- Check your work: After each subtraction, double-check your signs and combined terms. A small error early on can throw off the entire problem.
- Practice, practice, practice: The more you work through these problems, the more comfortable you’ll become with the process.
By understanding the underlying principles and meticulously working through each step, you can conquer polynomial division. Remember, everyone makes mistakes – the important thing is to learn from them and keep going!
Final Thoughts: Embracing the Learning Process
Math isn't about being perfect; it's about the journey of understanding. We all stumble sometimes, and that's okay! By analyzing Randy's mistake, we’ve reinforced the importance of careful steps and a solid grasp of the fundamentals. So, the next time you're faced with a polynomial division, take a deep breath, break it down step by step, and remember what we’ve learned today. You got this, guys! Keep up the great work, and happy calculating!
