Synthetic Division Practice Worksheet Answers Explained

gpTech

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11-16-2024

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Synthetic division is a method used for dividing polynomials, and it offers a streamlined alternative to traditional long division. It is especially useful when dividing by linear factors. In this article, we'll explore synthetic division, its application, and go through a practice worksheet with answers explained in detail. By the end, you should have a clearer understanding of how to perform synthetic division effectively! 🧠

What is Synthetic Division? 🤔

Synthetic division is a simplified form of polynomial division that is particularly helpful when dividing by a linear polynomial of the form (x - c). Unlike long division, synthetic division uses only the coefficients of the polynomials, making the process faster and more efficient.

Steps for Synthetic Division

To carry out synthetic division, follow these steps:

1. Identify the coefficients of the polynomial to be divided.
2. Set up the synthetic division using the root of the divisor (if dividing by (x - c), use (c)).
3. Carry down the leading coefficient.
4. Multiply the divisor root by the number directly above it and write the result under the next coefficient.
5. Add the columns and repeat the process until you have processed all coefficients.
6. The last row will represent the coefficients of the quotient, and the final value will represent the remainder.

Example Problem 📝

Let's go through an example problem to better understand synthetic division. Consider the division of (2x^3 - 6x^2 + 2x - 4) by (x - 3).

Step-by-Step Solution

Step 1: Identify the coefficients of the dividend:

• Coefficients: 2, -6, 2, -4

Step 2: Set up synthetic division:

• Using (c = 3):

3 |  2  -6   2  -4

Step 3: Carry down the leading coefficient:

3 |  2  -6   2  -4
   |      
   |  2

Step 4: Multiply and add:

3 |  2  -6   2  -4
   |      6  6
   |  2   0  8

Step 5: The result:

• Quotient: (2x^2 + 0x + 8) or just (2x^2 + 8)
• Remainder: (-4)

So, we can express the result as:

[ 2x^3 - 6x^2 + 2x - 4 = (x - 3)(2x^2 + 8) - 4 ]

Practice Worksheet 🏫

Let's create a simple practice worksheet for you to try!

Worksheet Problems:

1. Divide (4x^3 + 5x^2 - 3x + 1) by (x - 2).
2. Divide (x^4 - 4x^3 + 6x^2 - 4x + 1) by (x - 1).
3. Divide (3x^2 + 10x + 3) by (x + 1).

Answers Explained

Now, let's go through the answers for each problem on the practice worksheet.

Problem 1

Divide: (4x^3 + 5x^2 - 3x + 1) by (x - 2).

Solution:

1. Coefficients: (4, 5, -3, 1)
2. Set up:

2 |  4  5  -3  1

3. Carry down (4):

2 |  4  5  -3  1
   |     
   |  4

4. Multiply and add:

2 |  4  5  -3  1
   |     8  10
   |  4  13  7

• Quotient: (4x^2 + 13x + 7)
• Remainder: (15)

Problem 2

Divide: (x^4 - 4x^3 + 6x^2 - 4x + 1) by (x - 1).

Solution:

1. Coefficients: (1, -4, 6, -4, 1)
2. Set up:

1 |  1  -4  6  -4  1

3. Carry down (1):

1 |  1  -4  6  -4  1
   |      
   |  1

4. Multiply and add:

1 |  1  -4  6  -4  1
   |     -3  3
   |  1  -3  9  5

• Quotient: (x^3 - 3x^2 + 9x + 5)
• Remainder: (6)

Problem 3

Divide: (3x^2 + 10x + 3) by (x + 1).

Solution:

1. Coefficients: (3, 10, 3)
2. Set up:

-1 |  3  10  3

3. Carry down (3):

-1 |  3  10  3
    |      
    |  3

4. Multiply and add:

-1 |  3  10  3
    |    -3  -7
    |  3   7  -4

• Quotient: (3x + 7)
• Remainder: (-4)

Conclusion

Synthetic division is a powerful tool in polynomial algebra that simplifies the division process. By following the steps provided and practicing with various problems, you can master this technique. If you have any doubts, revisit the examples provided or try additional practice problems to reinforce your understanding. Happy dividing! 🎉