Polynomial Division: Find The Quotient!

Oct 22, 2025 by SLV Team 40 views

Hey guys! Today, we're diving into a fun problem involving polynomial division. Specifically, we need to figure out what you get when you divide the polynomial 2x⁴ - 3x³ - 3x² + 7x - 3 by x² - 2x + 1. Sounds like a mouthful, right? But don't worry, we'll break it down step by step so it's super easy to follow. So, grab your pencils, and let's get started!

Understanding Polynomial Division

Before we jump into the problem, let's quickly recap what polynomial division actually is. Think of it like regular long division, but instead of numbers, we're dealing with polynomials. The goal is the same: to find out how many times one polynomial (the divisor) fits into another polynomial (the dividend) and what's left over (the remainder).

Why is this important? Well, polynomial division is a fundamental tool in algebra and calculus. It helps us simplify complex expressions, factor polynomials, and solve equations. Plus, it's just a cool skill to have under your belt!

Setting Up the Division

Okay, let's get back to our problem. We're dividing 2x⁴ - 3x³ - 3x² + 7x - 3 (the dividend) by x² - 2x + 1 (the divisor). We'll set it up just like long division:

x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3

Now, we're ready to start dividing!

Step-by-Step Division

Here's how we'll tackle this division step by step:

Divide the first term: Look at the first terms of both polynomials. What do we need to multiply x² by to get 2x⁴? The answer is 2x². Write this above the line, aligned with the x² term in the dividend.
```
    2x²
```

x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 ```

Multiply: Multiply the entire divisor (x² - 2x + 1) by 2x²:

2x² * (x² - 2x + 1) = 2x⁴ - 4x³ + 2x²
Subtract: Subtract the result from the dividend:
```
    2x²
```

x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) ------------------- x³ - 5x² + 7x ```

Bring down the next term: Bring down the next term from the dividend (+7x):
```
    2x²
```

x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) ------------------- x³ - 5x² + 7x - 3 ```

Repeat: Now, repeat the process with the new polynomial x³ - 5x² + 7x - 3. What do we need to multiply x² by to get x³? The answer is x. Write this above the line, aligned with the x term.
```
    2x² + x
```

x² - 2x + 1 | 2x⁴ - 3x³ - 3x² + 7x - 3 -(2x⁴ - 4x³ + 2x²) ------------------- x³ - 5x² + 7x - 3 ```

Multiply: Multiply the divisor (x² - 2x + 1) by x:

x * (x² - 2x + 1) = x³ - 2x² + x
Subtract: 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) ---------------- -3x² + 6x - 3 ```

Repeat Again: Repeat the process one last time with the new polynomial -3x² + 6x - 3. What do we need to multiply x² by to get -3x²? The answer is -3. Write this above the line.
```
    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: Multiply the divisor (x² - 2x + 1) by -3:

-3 * (x² - 2x + 1) = -3x² + 6x - 3
Subtract: 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) ---------------- 0 ```

The Quotient

We've reached the end of the division, and the remainder is 0. That means the quotient is the polynomial we wrote above the line: 2x² + x - 3.

Therefore, the quotient of (2x⁴ - 3x³ - 3x² + 7x - 3) ÷ (x² - 2x + 1) is 2x² + x - 3.

Factoring and Verification

Just to be extra sure, let's quickly verify our answer. We can do this by multiplying the quotient by the divisor. If we did everything correctly, we should get back the original dividend.

(x² - 2x + 1) * (2x² + x - 3) = ?

Let's expand this:

x² * (2x² + x - 3) = 2x⁴ + x³ - 3x² -2x * (2x² + x - 3) = -4x³ - 2x² + 6x 1 * (2x² + x - 3) = 2x² + x - 3

Now, let's add these up:

2x⁴ + x³ - 3x² - 4x³ - 2x² + 6x + 2x² + x - 3 = 2x⁴ - 3x³ - 3x² + 7x - 3

And there you have it! We got back the original dividend, so we know our quotient is correct.

Also, notice that the divisor x² - 2x + 1 can be factored as (x - 1)². This is a perfect square trinomial, which makes the division a bit cleaner.

Answering the Question

So, to directly answer the question: The quotient of (2x⁴ - 3x³ - 3x² + 7x - 3) ÷ (x² - 2x + 1) is 2x² + x - 3.

Key Takeaways

Polynomial division is similar to long division with numbers.
The goal is to find the quotient and remainder.
Verifying your answer by multiplying the quotient and divisor is a good practice.
Factoring can sometimes simplify the process.

Practice Makes Perfect

Polynomial division might seem tricky at first, but with practice, you'll become a pro. Try working through some more examples to solidify your understanding. You can find plenty of practice problems online or in textbooks.

And that's it for today, guys! I hope this explanation was helpful. Keep practicing, and you'll be dividing polynomials like a boss in no time!

If you found this helpful, feel free to share it with your friends. Happy dividing!