Polynomial Division: Finding The Quotient Of Two Expressions

Understanding polynomial division is crucial in algebra. In this article, we'll dive deep into the process of dividing polynomials, specifically focusing on how to find the quotient when a higher-degree polynomial is divided by a lower-degree polynomial. We will use the example of dividing $(x^5 - 3x^3 - 3x^2 - 10x + 15)$ by $(x^2 - 5)$ to illustrate the method. Let's embark on this mathematical journey together!

Understanding Polynomial Division

Polynomial division is an extension of the familiar long division method used with numbers. In the context of polynomials, this process allows us to divide a polynomial (the dividend) by another polynomial (the divisor), resulting in a quotient and a remainder. The key idea is to systematically eliminate terms in the dividend until we reach a remainder that has a lower degree than the divisor. Before diving into the specifics of our example, let's clarify some key concepts. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples include $x^2 + 3x - 2$ and $5x^4 - 7x + 1$. When we divide polynomials, we're essentially trying to find out how many times the divisor "fits" into the dividend. This "fitting" is represented by the quotient, which is also a polynomial. The remainder is what's left over after the division, and its degree must be less than the degree of the divisor. The degree of a polynomial is the highest power of the variable in the polynomial. For instance, the degree of $x^3 - 2x + 1$ is 3. Mastering polynomial division opens doors to simplifying complex expressions, solving equations, and understanding the behavior of polynomial functions. It is an essential tool in various fields, including engineering, computer science, and economics, where polynomial models are frequently used to represent real-world phenomena. In the following sections, we will apply these concepts to our specific example, breaking down each step to ensure a clear understanding of the process. Let's proceed to the next section where we set up the long division for our problem.

Setting Up the Long Division

Now, let's apply polynomial division to our specific problem: dividing $(x^5 - 3x^3 - 3x^2 - 10x + 15)$ by $(x^2 - 5)$. The first step in any long division problem, whether with numbers or polynomials, is to set up the division correctly. We write the dividend, which is $(x^5 - 3x^3 - 3x^2 - 10x + 15)$, inside the long division symbol, and the divisor, $(x^2 - 5)$, outside. It's crucial to ensure that the polynomials are written in descending order of their exponents. This means starting with the term with the highest power of $x$ and proceeding to the constant term. In our dividend, we have $x^5$, $x^3$, $x^2$, $x$, and a constant term. However, notice that the $x^4$ term is missing. When setting up long division with polynomials, it's essential to include placeholders for any missing terms. We do this by adding terms with a coefficient of 0. In this case, we'll include $0x^4$ in the dividend to maintain the proper order and alignment of terms during the division process. This ensures that we don't make mistakes when subtracting terms later on. So, our dividend will be rewritten as $(x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15)$. This might seem like a small detail, but it's a crucial step in preventing errors. Now, we can write the long division setup as follows:

        ________________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15

With the problem set up correctly, we are ready to begin the actual division process. The next step involves focusing on the leading terms of the dividend and the divisor to determine the first term of the quotient. This is where we start to systematically eliminate terms in the dividend, working towards finding the final quotient and remainder. Let's move on to the next section to see how this process unfolds. Remember, careful setup is half the battle in long division, so let's proceed with confidence!

Performing the Division

Now that we have the long division set up correctly, let's perform the division. Our goal is to find the polynomial that, when multiplied by $(x^2 - 5)$, gives us $(x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15)$. We start by looking at the leading terms of the dividend ($x^5$) and the divisor ($x^2$). We ask ourselves: what do we need to multiply $x^2$ by to get $x^5$? The answer is $x^3$. So, $x^3$ is the first term of our quotient. We write $x^3$ above the $x^3$ term in the dividend (this is why aligning terms by their powers is so important!). Next, we multiply the entire divisor $(x^2 - 5)$ by $x^3$:

$x^3 * (x^2 - 5) = x^5 - 5x^3$

We write this result below the corresponding terms in the dividend:

        x^3 ____________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
        x^5       - 5x^3

Now, we subtract the expression we just wrote from the dividend. Remember to distribute the negative sign:

$(x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15) - (x^5 - 5x^3) = 0x^5 + 0x^4 + 2x^3 - 3x^2 - 10x + 15$

We bring down the remaining terms to get our new dividend: $2x^3 - 3x^2 - 10x + 15$. Now, we repeat the process. We look at the leading term of the new dividend ($2x^3$) and the leading term of the divisor ($x^2$). What do we need to multiply $x^2$ by to get $2x^3$? The answer is $2x$. So, $2x$ is the next term in our quotient. We write $+2x$ next to $x^3$ in the quotient. Multiply the divisor by $2x$:

$2x * (x^2 - 5) = 2x^3 - 10x$

Write this below the corresponding terms in the new dividend:

        x^3 + 2x____________
x^2 - 5 | 2x^3 - 3x^2 - 10x + 15
        2x^3       - 10x

Subtract again:

$(2x^3 - 3x^2 - 10x + 15) - (2x^3 - 10x) = -3x^2 + 15$

We repeat the process one more time. What do we multiply $x^2$ by to get $-3x^2$? The answer is -3. Write -3 in the quotient. Multiply the divisor by -3:

$-3 * (x^2 - 5) = -3x^2 + 15$

Write this below the new dividend:

        x^3 + 2x - 3
x^2 - 5 | -3x^2 + 15
        -3x^2 + 15

Subtract one last time:

$(-3x^2 + 15) - (-3x^2 + 15) = 0$

We have a remainder of 0. This means that $(x^2 - 5)$ divides evenly into $(x^5 - 3x^3 - 3x^2 - 10x + 15)$.

Determining the Quotient

After performing the polynomial long division, we've arrived at a clear result. The quotient is the polynomial we obtained above the division symbol. In our case, the quotient is $x^3 + 2x - 3$. This means that when you divide $(x^5 - 3x^3 - 3x^2 - 10x + 15)$ by $(x^2 - 5)$, you get $x^3 + 2x - 3$ with no remainder. To double-check our answer, we can multiply the quotient $(x^3 + 2x - 3)$ by the divisor $(x^2 - 5)$ and see if we get back the original dividend $(x^5 - 3x^3 - 3x^2 - 10x + 15)$. Let's perform this check:

$(x^3 + 2x - 3) * (x^2 - 5) = x^3 * (x^2 - 5) + 2x * (x^2 - 5) - 3 * (x^2 - 5)$

$= (x^5 - 5x^3) + (2x^3 - 10x) + (-3x^2 + 15)$

$= x^5 - 5x^3 + 2x^3 - 10x - 3x^2 + 15$

$= x^5 - 3x^3 - 3x^2 - 10x + 15$

As we can see, multiplying the quotient by the divisor gives us the original dividend, confirming that our division is correct. Therefore, the quotient of $(x^5 - 3x^3 - 3x^2 - 10x + 15)$ and $(x^2 - 5)$ is indeed $x^3 + 2x - 3$. This result is a polynomial, as expected, and it represents the result of the division. Understanding how to determine the quotient in polynomial division is a fundamental skill in algebra, enabling us to simplify expressions, solve equations, and analyze polynomial functions more effectively. In the next section, we will summarize the steps we've taken and highlight the key takeaways from this example.

Conclusion

In this article, we've walked through the process of polynomial division using a specific example: dividing $(x^5 - 3x^3 - 3x^2 - 10x + 15)$ by $(x^2 - 5)$. We started by understanding the basic concepts of polynomial division, including the roles of the dividend, divisor, quotient, and remainder. We emphasized the importance of setting up the long division correctly, including the use of placeholders for missing terms to ensure accurate alignment during the division process. Next, we meticulously performed the long division, step by step. We focused on the leading terms of the dividend and divisor to determine each term of the quotient. We multiplied the divisor by the term we added to the quotient and subtracted the result from the dividend. We repeated this process until we reached a remainder with a degree lower than the divisor. In our example, we found that the quotient is $x^3 + 2x - 3$ and the remainder is 0, indicating that $(x^2 - 5)$ divides evenly into $(x^5 - 3x^3 - 3x^2 - 10x + 15)$. To verify our result, we multiplied the quotient by the divisor and confirmed that we obtained the original dividend. This serves as a crucial step in ensuring the accuracy of our calculations. Polynomial division is a valuable skill in algebra, providing a way to simplify complex expressions and solve equations involving polynomials. It's also a foundational concept for further studies in mathematics, such as calculus and abstract algebra. By mastering polynomial division, you gain a deeper understanding of polynomial functions and their behavior. Remember, the key to success in polynomial division lies in careful setup, methodical execution, and a clear understanding of the underlying concepts. Practice makes perfect, so try working through additional examples to solidify your skills. For further learning on polynomials, visit Khan Academy's Polynomial Arithmetic section.