Polynomial Division Of (16x^4 - 24x^3 + 3) By (4x^2 + 3)

Polynomial division can often seem daunting, but with a systematic approach, it becomes a manageable task. In this comprehensive guide, we will dissect the process of dividing the polynomial $\frac{16x^4 - 24x^3 + 3}{4x^2 + 3}$. This detailed walkthrough aims to equip you with the skills necessary to tackle similar polynomial division problems with confidence. Understanding the mechanics of polynomial division not only enhances your algebraic proficiency but also lays a strong foundation for advanced mathematical concepts.

Understanding Polynomial Division

Before diving into the specifics of our problem, let's clarify the fundamental concepts of polynomial division. Polynomial division is an algebraic technique used to divide a polynomial by another polynomial of lower or equal degree. This process is akin to long division with numbers, but instead of digits, we're dealing with algebraic terms. Polynomial division is a critical skill in algebra, allowing us to simplify complex expressions, factor polynomials, and solve equations. The dividend is the polynomial being divided (in our case, $16x^4 - 24x^3 + 3$), and the divisor is the polynomial we are dividing by (here, $4x^2 + 3$). The result of the division gives us a quotient and a remainder. The quotient is the polynomial that results from the division, while the remainder is what is left over after the division, if any. Mastering this technique opens doors to more advanced algebraic manipulations and problem-solving strategies. Familiarity with polynomial division is essential for simplifying expressions, solving equations, and understanding the behavior of polynomial functions.

Setting Up the Long Division

To begin, we set up the long division similar to how we would with numerical long division. The dividend ($16x^4 - 24x^3 + 3$) goes inside the division symbol, and the divisor ($4x^2 + 3$) goes outside. It's crucial to include placeholders for any missing terms in the dividend. In our case, we have terms for $x^4$ and $x^3$, but we're missing terms for $x^2$ and $x$. To account for these, we rewrite the dividend as $16x^4 - 24x^3 + 0x^2 + 0x + 3$. These placeholders ensure that we align like terms properly during the division process. Proper alignment is essential for accurate calculations and helps prevent errors. Setting up the division correctly from the start streamlines the entire process and makes it easier to follow each step. This initial setup is a cornerstone of successful polynomial division, providing a clear framework for the subsequent steps. By including placeholders, we maintain the structure of the polynomial and avoid potential confusion.

Step-by-Step Division Process

Step 1: Divide the Leading Terms

Our first step is to divide the leading term of the dividend ($16x^4$) by the leading term of the divisor ($4x^2$). This gives us $\frac{16x^4}{4x^2} = 4x^2$. This result, $4x^2$, is the first term of our quotient. We write this term above the division symbol, aligning it with the $x^2$ term. Dividing the leading terms is the initial move in the division process, setting the stage for subsequent steps. This step determines the highest degree term in the quotient and directs the flow of the division. Accuracy in this initial step is critical, as it influences the rest of the calculation. By focusing on the leading terms, we simplify the problem and break it down into manageable parts.

Step 2: Multiply the Quotient Term by the Divisor

Next, we multiply the first term of the quotient ($4x^2$) by the entire divisor ($4x^2 + 3$). This yields $4x^2(4x^2 + 3) = 16x^4 + 12x^2$. We write this result below the dividend, aligning like terms. Multiplying the quotient term by the divisor is a critical step in the long division process. This multiplication allows us to determine what portion of the dividend is accounted for by the current term in the quotient. Aligning like terms ensures that the subsequent subtraction is performed accurately. This step reflects the distributive property of multiplication over addition and is fundamental to polynomial division.

Step 3: Subtract and Bring Down

We subtract the result from the corresponding terms in the dividend. So, we subtract $(16x^4 + 12x^2)$ from $(16x^4 - 24x^3 + 0x^2)$. This gives us $(16x^4 - 24x^3 + 0x^2) - (16x^4 + 12x^2) = -24x^3 - 12x^2$. Then, we bring down the next term from the dividend, which is $+0x$. This gives us $-24x^3 - 12x^2 + 0x$. Subtracting and bringing down terms is a core step in long division, mirroring the process used in numerical long division. This subtraction effectively reduces the degree of the dividend, moving us closer to the final quotient and remainder. Bringing down the next term ensures that all terms in the dividend are considered in the division process. This step combines arithmetic subtraction with algebraic manipulation, requiring careful attention to signs and like terms.

Step 4: Repeat the Process

Now, we repeat the process with the new polynomial $-24x^3 - 12x^2 + 0x$. We divide the leading term $-24x^3$ by the leading term of the divisor $4x^2$, which gives us $\frac{-24x^3}{4x^2} = -6x$. This is the next term in our quotient. We write $-6x$ in the quotient, aligning it with the $x$ term. Then, we multiply $-6x$ by the divisor $4x^2 + 3$, which results in $-6x(4x^2 + 3) = -24x^3 - 18x$. We write this below the current polynomial and subtract: $(-24x^3 - 12x^2 + 0x) - (-24x^3 - 18x) = -12x^2 + 18x$. We bring down the last term from the dividend, which is $+3$, giving us $-12x^2 + 18x + 3$. Repeating the process ensures that the division is carried out until the degree of the remaining polynomial is less than the degree of the divisor. This iterative approach is the heart of long division, allowing us to systematically break down the dividend. Each repetition refines the quotient and remainder, bringing us closer to the final solution. The cyclical nature of this step reinforces the algorithm and improves proficiency with practice.

Step 5: Final Division and Remainder

We repeat the division one more time. Divide the leading term $-12x^2$ by the leading term of the divisor $4x^2$, which gives us $\frac{-12x^2}{4x^2} = -3$. This is the last term in our quotient. We write $-3$ in the quotient. Multiply $-3$ by the divisor $4x^2 + 3$, which results in $-3(4x^2 + 3) = -12x^2 - 9$. We subtract this from our current polynomial: $(-12x^2 + 18x + 3) - (-12x^2 - 9) = 18x + 12$. The degree of the remaining polynomial $18x + 12$ is less than the degree of the divisor $4x^2 + 3$, so this is our remainder. Final division determines the last term of the quotient, and the remainder represents what is left over after the division process. The remainder's degree being less than the divisor's degree signals the termination of the division. This step encapsulates the culmination of the entire process, resulting in the complete quotient and remainder.

Expressing the Result

The quotient we found is $4x^2 - 6x - 3$, and the remainder is $18x + 12$. Therefore, we can express the result of the division as: $\frac{16x^4 - 24x^3 + 3}{4x^2 + 3} = 4x^2 - 6x - 3 + \frac{18x + 12}{4x^2 + 3}$. Expressing the result in the standard form of quotient plus remainder over divisor is the final step in presenting the solution. This representation clearly delineates the quotient and the fractional remainder. The final expression provides a complete and concise answer to the polynomial division problem.

Final Answer: $\frac{16 x^4-24 x^3+3}{4 x^2+3} = 4x^2-6x-3+\frac{18x+12}{4x^2+3}$

In conclusion, mastering polynomial division involves a systematic approach, including setting up the problem correctly, performing the division step-by-step, and expressing the result in the appropriate form. With practice, you can become proficient in this essential algebraic skill. This detailed guide has provided a thorough walkthrough of dividing $\frac{16x^4 - 24x^3 + 3}{4x^2 + 3}$, equipping you with the knowledge and confidence to tackle similar problems effectively. Polynomial division, while intricate, becomes manageable with a clear understanding of the process and consistent practice.