Long Division: Finding Quotient & Remainder Explained

Hey everyone, let's dive into the fascinating world of long division! Today, we're going to break down how to find the quotient and remainder when dividing the polynomial $x^2 + 8x + 17$ by the binomial $x + 3$. Don't worry if this sounds intimidating at first; I'll walk you through it step-by-step. Long division, in essence, is a method for dividing a polynomial by another polynomial of equal or lesser degree. It's similar to the long division you learned in elementary school, but we're working with variables and exponents now. The key is to take it slow, focus on one term at a time, and remember the basic principles of division: Divide, Multiply, Subtract, and Bring Down. That's our mantra, okay? We will also use remainder theorem and other techniques to verify the solution. By the end of this, you will have a solid understanding of how to perform polynomial long division and find both the quotient and the remainder. Let's get started, shall we?

Setting Up the Long Division Problem

Alright, first things first, let's set up our long division problem. We're going to write it out like this:

x + 3 | x^2 + 8x + 17

The polynomial $x^2 + 8x + 17$ goes inside the division symbol, and the binomial $x + 3$ goes outside. This is exactly like how you'd set up a regular long division problem with numbers. Make sure that both the dividend (the polynomial inside the division symbol) and the divisor (the polynomial outside the division symbol) are written in standard form, meaning the terms are arranged in descending order of their exponents. In our case, they already are. Polynomial long division is a fundamental technique in algebra, used not just to solve problems like this, but also as a stepping stone to understanding more complex concepts. It allows us to break down a larger polynomial expression into simpler parts, helping us to identify its roots, factorize it, and sketch its graph. It is also an important element for factor theorem. We will carefully and patiently work through each step to ensure we get to the correct solution. Remember, practice is key, so the more you do these problems, the more comfortable you'll become. So, without further ado, let's begin the division process.

The Division Process: Step-by-Step

Now, let's get into the actual division. We'll follow the Divide, Multiply, Subtract, Bring Down steps. This process is repeated until we can no longer divide. Let's start:

Step 1: Divide

Focus on the first term of the dividend ($x^2$) and the first term of the divisor ($x$). Ask yourself: What do I need to multiply $x$ by to get $x^2$? The answer is $x$. So, we write $x$ on top, above the $8x$ term.

      x
x + 3 | x^2 + 8x + 17

Step 2: Multiply

Next, we multiply the $x$ we just wrote on top by the entire divisor, $(x + 3)$.

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

Write this result under the dividend.

      x
x + 3 | x^2 + 8x + 17
        x^2 + 3x

Step 3: Subtract

Now, subtract the result from the dividend. Remember to subtract the entire expression, $(x^2 + 3x)$.

      x
x + 3 | x^2 + 8x + 17
        x^2 + 3x
        ------

When we subtract, we change the signs of the terms in the second row and combine like terms:

$(x^2 - x^2) + (8x - 3x) + 17 = 5x + 17$

Write the result below.

      x
x + 3 | x^2 + 8x + 17
        x^2 + 3x
        ------
            5x + 17

Step 4: Bring Down

There's nothing to bring down in this step, as we've already dealt with the $x^2$ and $8x$ terms. Since we have a remainder, we'll continue the process with the remaining terms.

Repeat the Process

Now, repeat the Divide, Multiply, Subtract steps with the new polynomial, $5x + 17$.

• Divide: $5x / x = 5$. Write $+5$ on top, next to the $x$.

        x + 5
  

x + 3 | x^2 + 8x + 17 x^2 + 3x ------ 5x + 17 ```

• Multiply: $5 * (x + 3) = 5x + 15$. Write this result under $5x + 17$.

        x + 5
  

x + 3 | x^2 + 8x + 17 x^2 + 3x ------ 5x + 17 5x + 15 ```

• Subtract: $(5x + 17) - (5x + 15) = 2$. Write the result below.

        x + 5
  

x + 3 | x^2 + 8x + 17 x^2 + 3x ------ 5x + 17 5x + 15 ------ 2 ```

Step 5: The Remainder

We are left with 2, and since the degree of 2 (which is 0) is less than the degree of the divisor ($x + 3$, which is 1), we cannot divide further. This means our remainder is 2.

Identifying the Quotient and Remainder

Alright, we've successfully completed the long division. Now, let's identify the quotient and the remainder.

• Quotient: The quotient is the expression on top of the division symbol. In our case, the quotient is $x + 5$.
• Remainder: The remainder is what's left over after the final subtraction. Here, the remainder is $2$.

So, when we divide $x^2 + 8x + 17$ by $x + 3$, we get a quotient of $x + 5$ and a remainder of $2$. This can be expressed as:

rac{x^2 + 8x + 17}{x + 3} = x + 5 + rac{2}{x + 3}

This is a super important concept because it allows us to analyze the behavior of the original polynomial. For example, knowing the quotient and remainder can help us graph the polynomial or determine its end behavior. And it also lays the groundwork for understanding more advanced mathematical concepts like partial fraction decomposition. This is a very powerful tool! So, keep up the great work, you guys!

Verifying the Solution (Optional)

We can always verify our solution by multiplying the quotient by the divisor and adding the remainder. If we get back the original dividend, we know we've done it correctly.

$(x + 5)(x + 3) + 2 = x^2 + 3x + 5x + 15 + 2 = x^2 + 8x + 17$

Since this matches our original dividend, we know our answer is correct! Pretty neat, huh? Also, we can also use remainder theorem for verification.

Conclusion

And that, my friends, is how you perform long division with polynomials! We have successfully found both the quotient and the remainder of the division. This process is a fundamental skill in algebra and is essential for understanding more complex topics in mathematics. Remember, the key is to take it slow, focus on each step, and practice as much as possible. With practice, you'll become a pro at long division in no time. If you have any questions, feel free to ask in the comments below. Keep practicing and keep learning, and you'll do great! And that's all, folks!