Identify The Quotient And The Remainder In Polynomial Division

#h1 Identify the Quotient and the Remainder

In the realm of polynomial division, understanding how to identify the quotient and the remainder is a fundamental skill. It allows us to break down complex polynomial expressions into simpler, more manageable forms. This article will delve into the process of polynomial division, specifically focusing on identifying the quotient and the remainder when dividing $(24x^3 - 14x^2 + 20x + 6)$ by $(4x^2 - 3x + 5)$. We will explore the steps involved in long division, highlighting how each term is carefully manipulated to arrive at the final quotient and remainder. The ability to perform polynomial division is crucial in various areas of mathematics, including algebra, calculus, and more advanced topics. It's not just about following a set of steps; it's about grasping the underlying principles that govern how polynomials interact with each other. Let's embark on this journey to master the art of polynomial division and confidently identify the quotient and the remainder.

Understanding Polynomial Division

Before diving into the specifics of the given problem, it's essential to grasp the concept of polynomial division. Polynomial division is analogous to long division with numbers, but instead of digits, we're dealing with terms containing variables and exponents. The goal is the same: to divide one polynomial (the dividend) by another (the divisor) to obtain a quotient and a remainder. The quotient represents the whole number of times the divisor goes into the dividend, while the remainder is the part that's left over. In our case, the dividend is $(24x^3 - 14x^2 + 20x + 6)$, and the divisor is $(4x^2 - 3x + 5)$. Our objective is to find the polynomials $Q$ (the quotient) and $R$ (the remainder) such that:

$(24x^3 - 14x^2 + 20x + 6) ext{ divided by } (4x^2 - 3x + 5) = Q + \frac{R}{4x^2 - 3x + 5}$

This equation represents the fundamental relationship in polynomial division. The dividend is equal to the divisor multiplied by the quotient, plus the remainder. In other words, if we were to multiply the quotient by the divisor and add the remainder, we should arrive back at our original dividend. This relationship serves as a crucial check for the accuracy of our division process. The remainder will always have a degree less than the divisor, which is a key characteristic to keep in mind as we perform the division. The process involves a series of steps that carefully subtract multiples of the divisor from the dividend until we reach a remainder that cannot be further divided. The quotient is built up term by term as we go through these steps. Understanding the rationale behind each step is vital for mastering the technique of polynomial division. With this foundational understanding, we can now proceed to tackle the specific problem at hand and methodically determine the quotient and remainder.

Performing Long Division with Polynomials

To identify the quotient and remainder, we'll use the method of long division for polynomials. This method systematically breaks down the division process into manageable steps. Let's set up the long division as follows:

 4x^2 - 3x + 5 | 24x^3 - 14x^2 + 20x + 6

First, we focus on the leading terms of both the dividend $(24x^3)$ and the divisor $(4x^2)$. We ask ourselves: what do we need to multiply $4x^2$ by to get $24x^3$? The answer is $6x$. This becomes the first term of our quotient. We write $6x$ above the $20x$ term in the dividend, aligning terms with the same degree.

 6x
 4x^2 - 3x + 5 | 24x^3 - 14x^2 + 20x + 6

Next, we multiply the entire divisor $(4x^2 - 3x + 5)$ by $6x$:

$6x * (4x^2 - 3x + 5) = 24x^3 - 18x^2 + 30x$

We write this result below the dividend, aligning like terms:

 6x
 4x^2 - 3x + 5 | 24x^3 - 14x^2 + 20x + 6
 24x^3 - 18x^2 + 30x

Now, we subtract the result from the dividend. Remember to change the signs of the terms being subtracted:

 6x
 4x^2 - 3x + 5 | 24x^3 - 14x^2 + 20x + 6
 - (24x^3 - 18x^2 + 30x)
 -------------------------
 4x^2 - 10x + 6

We bring down the next term from the dividend, which is $+6$. This gives us a new polynomial to work with: $4x^2 - 10x + 6$. Now, we repeat the process. We ask: what do we need to multiply $4x^2$ by to get $4x^2$? The answer is $1$. This becomes the next term in our quotient. We write $+1$ next to the $6x$ in the quotient.

 6x + 1
 4x^2 - 3x + 5 | 24x^3 - 14x^2 + 20x + 6
 - (24x^3 - 18x^2 + 30x)
 -------------------------
 4x^2 - 10x + 6

We multiply the divisor $(4x^2 - 3x + 5)$ by $1$:

$1 * (4x^2 - 3x + 5) = 4x^2 - 3x + 5$

We write this below $4x^2 - 10x + 6$:

 6x + 1
 4x^2 - 3x + 5 | 24x^3 - 14x^2 + 20x + 6
 - (24x^3 - 18x^2 + 30x)
 -------------------------
 4x^2 - 10x + 6
 4x^2 - 3x + 5

Subtract again, changing the signs of the terms being subtracted:

 6x + 1
 4x^2 - 3x + 5 | 24x^3 - 14x^2 + 20x + 6
 - (24x^3 - 18x^2 + 30x)
 -------------------------
 4x^2 - 10x + 6
 - (4x^2 - 3x + 5)
 ----------------
 -7x + 1

The result of the subtraction is $-7x + 1$. This is our remainder because its degree (1) is less than the degree of the divisor (2). We have now completed the long division process. This methodical approach, breaking down the problem into smaller steps, is crucial for accurate polynomial division. By carefully tracking each term and sign, we can confidently arrive at the correct quotient and remainder.

Identifying the Quotient and Remainder

After performing the long division, we can now clearly identify the quotient and the remainder. From the steps above, we found the quotient, denoted as $Q$, to be $6x + 1$. This represents the polynomial we obtained by dividing the leading terms at each stage of the long division process. The remainder, denoted as $R$, is the final polynomial left after the last subtraction, which is $-7x + 1$. This remainder has a lower degree than the divisor, as expected, indicating that we cannot divide further. Thus, we have:

• $Q = 6x + 1$
• $R = -7x + 1$

Therefore, the result of the polynomial division can be expressed as:

$(24x^3 - 14x^2 + 20x + 6) ext{ divided by } (4x^2 - 3x + 5) = (6x + 1) + \frac{-7x + 1}{4x^2 - 3x + 5}$

This equation confirms that the dividend is equal to the divisor multiplied by the quotient, plus the remainder. In other words:

$(24x^3 - 14x^2 + 20x + 6) = (4x^2 - 3x + 5)(6x + 1) + (-7x + 1)$

This final equation serves as a check on our work. We can expand the right side of the equation and verify that it indeed equals the left side, the original dividend. This step of verification is crucial in ensuring the accuracy of our polynomial division. By meticulously identifying the quotient and remainder, we have successfully broken down a complex polynomial division problem into its fundamental components. This skill is not only essential for algebraic manipulations but also for deeper concepts in calculus and other advanced mathematical fields.

Verification and Conclusion

To ensure the accuracy of our results, let's verify our solution. We will multiply the quotient $(6x + 1)$ by the divisor $(4x^2 - 3x + 5)$ and then add the remainder $(-7x + 1)$. If our division is correct, this should give us the original dividend $(24x^3 - 14x^2 + 20x + 6)$.

First, let's multiply the quotient and the divisor:

$(6x + 1)(4x^2 - 3x + 5) = 6x(4x^2 - 3x + 5) + 1(4x^2 - 3x + 5)$

$= 24x^3 - 18x^2 + 30x + 4x^2 - 3x + 5$

Now, combine like terms:

$= 24x^3 - 14x^2 + 27x + 5$

Next, add the remainder $(-7x + 1)$ to the result:

$(24x^3 - 14x^2 + 27x + 5) + (-7x + 1) = 24x^3 - 14x^2 + 20x + 6$

As we can see, the result matches our original dividend $(24x^3 - 14x^2 + 20x + 6)$. This confirms that our quotient and remainder are correct. In conclusion, by performing long division, we have successfully identified the quotient $Q = 6x + 1$ and the remainder $R = -7x + 1$ when dividing $(24x^3 - 14x^2 + 20x + 6)$ by $(4x^2 - 3x + 5)$. This process demonstrates the power of polynomial division in simplifying complex expressions and provides a solid foundation for further algebraic manipulations and problem-solving. The ability to accurately divide polynomials is a crucial skill in mathematics, enabling us to solve equations, simplify expressions, and delve deeper into advanced mathematical concepts. With practice and a clear understanding of the underlying principles, polynomial division becomes a valuable tool in any mathematician's arsenal.