Synthetic Division: Find The Quotient

Let's dive into how to use synthetic division to solve the polynomial division problem: $(x^3 - x^2 - 17x - 15) \\div (x - 5)$. We'll break down each step and find the quotient.

Understanding Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear expression of the form $(x - a)$. It's quicker and more efficient than long division, especially when dealing with higher-degree polynomials. This method simplifies the division process by focusing on the coefficients of the polynomial and the constant term of the divisor.

Before we start, it’s good to know what to look for in the result. The result of synthetic division will give us the coefficients of the quotient and the remainder. Understanding this principle is important in applying the technique correctly and interpreting the results effectively. So, keep in mind as we proceed through the steps: our goal is to extract these key pieces of information efficiently.

Setting Up Synthetic Division

To set up the synthetic division, follow these steps:

1. Identify the coefficients of the polynomial we want to divide (dividend) and write them in order. Make sure to include a 0 for any missing terms. In our case, the polynomial is $x^3 - x^2 - 17x - 15$, so the coefficients are 1, -1, -17, and -15.
2. Determine the value of a from the divisor $(x - a)$. Here, the divisor is $(x - 5)$, so $a = 5$. This is the number we'll use in the synthetic division process.
3. Draw a horizontal line and write the coefficients of the dividend to the right of the vertical line. Place the a value (which is 5 in our case) to the left of the vertical line.

Now the setup should look something like this:

5 | 1 -1 -17 -15
  |________________

Performing the Synthetic Division

Now, let's perform the synthetic division step by step:

1. Bring down the first coefficient (which is 1 in our example) below the horizontal line.

5 | 1 -1 -17 -15
  |________________
  1

2. Multiply the value we just brought down (1) by the a value (5), and write the result (5) under the next coefficient (-1).

5 | 1 -1 -17 -15
  | 5
  |________________
  1

3. Add the numbers in the second column (-1 and 5) and write the sum (4) below the horizontal line.

5 | 1 -1 -17 -15
  | 5
  |________________
  1 4

4. Multiply the new value (4) by the a value (5), and write the result (20) under the next coefficient (-17).

5 | 1 -1 -17 -15
  | 5 20
  |________________
  1 4

5. Add the numbers in the third column (-17 and 20) and write the sum (3) below the horizontal line.

5 | 1 -1 -17 -15
  | 5 20
  |________________
  1 4 3

6. Multiply the new value (3) by the a value (5), and write the result (15) under the next coefficient (-15).

5 | 1 -1 -17 -15
  | 5 20 15
  |________________
  1 4 3

7. Add the numbers in the last column (-15 and 15) and write the sum (0) below the horizontal line. This is the remainder.

5 | 1 -1 -17 -15
  | 5 20 15
  |________________
  1 4 3 0

Interpreting the Result

The numbers below the horizontal line (1, 4, and 3) are the coefficients of the quotient. Since we started with a cubic polynomial ($x^3$) and divided by a linear term ($x - 5$), the quotient will be a quadratic polynomial. The last number (0) is the remainder.

Therefore, the quotient is $1x^2 + 4x + 3$, which simplifies to $x^2 + 4x + 3$. The remainder is 0, which means the division is exact.

Conclusion

Using synthetic division, we found that $(x^3 - x^2 - 17x - 15) \\div (x - 5) = x^2 + 4x + 3$ with no remainder. This matches option A. So, the correct answer is:

A. $x^2 + 4x + 3$

Synthetic division is a handy tool for simplifying polynomial division, and with practice, it can save you a lot of time and effort. Remember to set up the problem correctly, follow the steps carefully, and interpret the result accurately.

For further learning and more examples on synthetic division, you can visit Khan Academy's page on synthetic division. This resource provides additional explanations and practice problems to enhance your understanding.