Polynomial Division Explained: A Step-by-Step Guide

Welcome to the fascinating world of polynomial division! Today, we're going to tackle a specific problem: dividing the polynomial $10x^4 - 14x^3 - 10x^2 + 6x - 10$ by $x^3 - 3x^2 + x - 2$. Polynomial division might sound intimidating at first, but think of it like long division with numbers, just with more variables and exponents involved. It's a fundamental skill in algebra that helps us understand the relationships between polynomials, factor them, and solve complex equations. We'll break down this process step-by-step, ensuring that by the end, you'll feel confident in your ability to perform this type of division. So, grab a pen and paper, and let's dive in!

Understanding the Basics of Polynomial Division

Before we jump into our specific example, let's quickly recap what polynomial division is all about. Polynomial division is an algorithm that allows us to divide a polynomial (the dividend) by another polynomial (the divisor) of a lower or equal degree. The result of this division is a quotient polynomial and a remainder polynomial. The remainder's degree will always be less than the degree of the divisor. This process is incredibly useful in algebra for several reasons. It's a key step in factoring polynomials, finding roots (or zeros) of polynomial equations, and simplifying rational expressions. Think of it as the algebraic equivalent of breaking down a complex number into simpler parts. The process mirrors long division with numbers, where you repeatedly subtract multiples of the divisor from the dividend until you're left with a remainder that's smaller than the divisor. In the polynomial world, "smaller" means having a lower degree. We'll be using a method very similar to numerical long division, carefully aligning terms by their degree and systematically eliminating the leading terms of the dividend at each step.

Setting Up the Polynomial Long Division

To begin dividing $10x^4 - 14x^3 - 10x^2 + 6x - 10$ by $x^3 - 3x^2 + x - 2$, we set it up just like numerical long division. The dividend, $10x^4 - 14x^3 - 10x^2 + 6x - 10$, goes inside the division symbol, and the divisor, $x^3 - 3x^2 + x - 2$, goes outside. It's crucial to ensure that both the dividend and the divisor are written in descending order of powers of x. If any terms are missing (e.g., no $x^2$ term), you should include them with a coefficient of zero as a placeholder. In our case, both polynomials are complete and in the correct order, so we can proceed directly. We write it out as:

        _____________
x^3-3x^2+x-2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10

Our goal is to find out how many times the leading term of the divisor ($x^3$) fits into the leading term of the dividend ($10x^4$). This first step is critical because it determines the first term of our quotient. We'll then multiply this term by the entire divisor and subtract the result from the dividend, bringing down the next term. This process repeats until the degree of the remaining polynomial is less than the degree of the divisor. It's like peeling layers off an onion, each step simplifying the expression further. Remember to be meticulous with your signs during subtraction, as this is a common source of errors in polynomial long division. Each successful step brings us closer to the final quotient and remainder, revealing the underlying structure of the polynomials we're working with. It's a methodical process, and with practice, it becomes almost second nature.

Step 1: Divide the Leading Terms

Let's start by focusing on the leading terms: $10x^4$ from the dividend and $x^3$ from the divisor. To find the first term of our quotient, we ask: what do we need to multiply $x^3$ by to get $10x^4$? The answer is $10x$ ($10x^4 / x^3 = 10x$). So, $10x$ becomes the first term of our quotient, which we write above the division symbol, aligned with the $x^3$ term:

        10x ________
x^3-3x^2+x-2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10

Now, we take this $10x$ and multiply it by the entire divisor: $10x * (x^3 - 3x^2 + x - 2)$. This gives us $10x^4 - 30x^3 + 10x^2 - 20x$. We write this result below the dividend, aligning terms with the same powers:

        10x ________
x^3-3x^2+x-2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
              10x^4 - 30x^3 + 10x^2 - 20x

Next, we subtract this expression from the dividend. Remember to change the signs of each term in the expression you are subtracting:

        10x ________
x^3-3x^2+x-2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
            -(10x^4 - 30x^3 + 10x^2 - 20x)
              -------------------------

Performing the subtraction:

• $(10x^4) - (10x^4) = 0x^4$
• $(-14x^3) - (-30x^3) = -14x^3 + 30x^3 = 16x^3$
• $(-10x^2) - (10x^2) = -10x^2 - 10x^2 = -20x^2$
• $(6x) - (-20x) = 6x + 20x = 26x$

So, our result after subtraction is $16x^3 - 20x^2 + 26x$. We then bring down the next term from the original dividend, which is $-10$:

        10x ________
x^3-3x^2+x-2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
            -(10x^4 - 30x^3 + 10x^2 - 20x)
              -------------------------
                     16x^3 - 20x^2 + 26x - 10

This new polynomial, $16x^3 - 20x^2 + 26x - 10$, becomes our new dividend for the next step. This initial step is often the most crucial, setting the tone for the rest of the division. By correctly identifying the first term of the quotient and performing the subtraction accurately, we've significantly reduced the complexity of the original problem.

Step 2: Repeat the Process

Now, we repeat the entire process with our new dividend: $16x^3 - 20x^2 + 26x - 10$. We again focus on the leading terms: $16x^3$ from our current dividend and $x^3$ from the original divisor. To find the next term in our quotient, we ask: what do we multiply $x^3$ by to get $16x^3$? The answer is $16$. So, $16$ becomes the next term in our quotient:

        10x + 16 ____
x^3-3x^2+x-2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
            -(10x^4 - 30x^3 + 10x^2 - 20x)
              -------------------------
                     16x^3 - 20x^2 + 26x - 10

Now, we multiply this new term, $16$, by the entire divisor: $16 * (x^3 - 3x^2 + x - 2)$. This yields $16x^3 - 48x^2 + 16x - 32$. We write this below our current dividend, aligning terms:

        10x + 16 ____
x^3-3x^2+x-2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
            -(10x^4 - 30x^3 + 10x^2 - 20x)
              -------------------------
                     16x^3 - 20x^2 + 26x - 10
                     16x^3 - 48x^2 + 16x - 32

Once again, we subtract this expression from our current dividend. Remember to change the signs:

        10x + 16 ____
x^3-3x^2+x-2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
            -(10x^4 - 30x^3 + 10x^2 - 20x)
              -------------------------
                     16x^3 - 20x^2 + 26x - 10
                   -(16x^3 - 48x^2 + 16x - 32)
                     -------------------------

Performing the subtraction:

• $(16x^3) - (16x^3) = 0x^3$
• $(-20x^2) - (-48x^2) = -20x^2 + 48x^2 = 28x^2$
• $(26x) - (16x) = 10x$
• $(-10) - (-32) = -10 + 32 = 22$

The result of this subtraction is $28x^2 + 10x + 22$. This is our remainder because its degree (2) is less than the degree of the divisor ($x^3$, which has degree 3). We don't have any more terms to bring down from the original dividend, so we are finished!

The Final Result: Quotient and Remainder

After completing the polynomial long division, we have found our quotient and our remainder. The quotient is the polynomial written above the division symbol, and the remainder is the final polynomial we obtained after the last subtraction.

• Quotient: $10x + 16$
• Remainder: $28x^2 + 10x + 22$

We can express the result of the division in the form: Dividend = Divisor * Quotient + Remainder. So, in this case:

$10x^4 - 14x^3 - 10x^2 + 6x - 10 = (x^3 - 3x^2 + x - 2)(10x + 16) + (28x^2 + 10x + 22)$

Alternatively, we can write the result as:

$$\frac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2} = 10x + 16 + \frac{28x^2 + 10x + 22}{x^3 - 3x^2 + x - 2}$$

This form explicitly shows the quotient and the fractional part consisting of the remainder over the divisor. Mastering polynomial division is a significant step in algebra. It opens doors to understanding more complex mathematical concepts and problem-solving techniques. Practice with different polynomials is key to becoming proficient. Keep in mind the careful alignment of terms and meticulous attention to signs during subtraction.

Conclusion

We've successfully navigated the process of polynomial long division for the expression rac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2}. By systematically dividing the leading terms, multiplying, and subtracting, we arrived at a quotient of $10x + 16$ and a remainder of $28x^2 + 10x + 22$. This method, while requiring careful attention to detail, is a powerful tool for simplifying and analyzing polynomial expressions. It's a foundational technique that underpins many advanced algebraic concepts, from factoring polynomials to understanding rational functions. If you're looking to deepen your understanding of algebraic manipulations and theorems related to polynomials, exploring resources on the Remainder Theorem and the Factor Theorem can provide further insight. These theorems offer elegant shortcuts and connections related to polynomial division and roots. For further exploration and practice, you might find the resources at Khan Academy extremely helpful in solidifying your grasp on polynomial operations.