I. Introduction
Synthetic division is a method for performing polynomial division that is often used in algebra and other fields. It is a more efficient way of dividing a polynomial by a linear factor than other methods of polynomial division, such as long division. Knowing how to perform synthetic division can save time and make solving problems easier.
In this article, we will explore how to perform synthetic division step-by-step, common mistakes to avoid, how it compares to other methods of polynomial division, its real-world applications, and how to check for correctness.
II. How to Perform Synthetic Division
Synthetic division is a simple and straightforward process that involves dividing a polynomial by a linear factor. To perform synthetic division, you will need the following tools:
1. Division symbol
The division symbol is represented by a long, curved line that separates the dividend (the polynomial being divided) from the divisor (the linear factor).
2. Coefficients of the polynomial
The coefficients of the polynomial are the numbers that appear in front of each term of the polynomial. They are used in the division process.
3. Divisor
The divisor is a linear factor of the form (x – a), where a is a constant.
To perform synthetic division, follow these step-by-step instructions:
1. Organizing the problem
Write the polynomial to be divided in descending order of degree. For example, if the polynomial is 3x3 + 2x2 – 4x + 1, write it as:
3x3 + 2x2 – 4x + 1
Next, write the divisor in the form (x – a), where a is a constant. For example, if the divisor is (x – 2), write it as:
(x – 2)
Note that instead of using the subtraction symbol (-), we use addition (+) to negate the constant term of the divisor. Hence, (x – 2) becomes (x + (-2)).
2. Writing the coefficients of the polynomial
Write the coefficients of the polynomial in a row below it. For the above polynomial, the row of coefficients would be:
3 2 -4 1
3. Writing the divisor
Write the constant term of the divisor in a box to the left of the row of coefficients. For the above polynomial, the constant term of the divisor is -2, so it would be written as:
-2 | 3 2 -4 1
4. Performing the division process
The first step in the division process is to bring down the first coefficient of the polynomial into the first row of the solution.
-2 | 3 2 -4 1
3
Next, multiply the constant term of the divisor (-2) by the number in the first row of the solution (3), and write the result under the second coefficient of the polynomial.
-2 | 3 2 -4 1
3
-6
Add the second coefficient of the polynomial (-4) and the result of the multiplication (-6), and write the result under the third coefficient of the polynomial.
-2 | 3 2 -4 1
3
-6
-10
Repeat this process until all of the coefficients have been processed. The final row of the solution contains the coefficients of the quotient polynomial, and the constant term of the divisor represents the remainder.
-2 | 3 2 -4 1
3
-6
-10
1
The quotient polynomial is 3x2 + 4x – 5, and the remainder is 1.
D. Examples to Illustrate the Process
Let’s look at some examples to illustrate the process of synthetic division:
Example 1:
Divide 2x3 – 9x2 – 7x + 6 by (x – 3).
-3 | 2 -9 -7 6
2 -15 18
-9 24 -72
15 -48 -138
The quotient polynomial is 2x2 – 3x – 46, and the remainder is -138.
Example 2:
Divide 3x4 – 8x3 – 10x2 + 12x + 4 by (x + 2).
-2 | 3 -8 -10 12 4
3 -6 4 -20
-6 10 -28 56
4 4 -24 -44
The quotient polynomial is 3x3 – 6x2 + 4x – 22, and the remainder is -44.
III. Common Mistakes to Avoid While Performing Synthetic Division
Although synthetic division is a simple and straightforward process, there are some common mistakes that can be made while performing it. Here are some common mistakes to avoid:
A. Explanation of common mistakes made while performing synthetic division
1. Forgetting to write the divisor in the required (x – a) form: It is important to remember to write the divisor in the required form, with the subtraction symbol (-) replaced by addition (+) before starting the division process.
2. Miscalculating the first term: The first term is obtained by simply bringing down the first coefficient of the polynomial. This is a common error that can easily be avoided by double-checking the first term.
3. Incorrectly multiplying the constant term of the divisor: When multiplying the constant term of the divisor with the first coefficient of the polynomial, it is important to remember to use the correct sign. For example, if the constant term of the divisor is -2, remember to multiply by -3 if the first coefficient of the polynomial is positive.
B. Tips for avoiding common mistakes
1. Double-checking numbers: Before moving on to the next step, double-check the numbers to make sure they are correct.
2. Keeping all steps organized: It is important to keep all steps organized and clear to avoid making mistakes. This can be done simply by drawing a line to separate the numbers, and writing the results under the correct coefficients of the polynomial.
C. Practice problems for readers to test their understanding and avoid mistakes
Here are some practice problems for readers to test their understanding and avoid common mistakes:
1. Divide 4x3 + 2x2 – 8x + 6 by (x – 2).
2. Divide 2x4 + 3x3 – 5x2 + 9x – 6 by (x + 1).
IV. Comparison of Synthetic Division to Other Methods of Polynomial Division
There are other methods of polynomial division, such as long division and polynomial factorization, that can be used instead of synthetic division. However, synthetic division has some advantages over these methods.
A. Explanation of other methods of polynomial division
1. Long division: Long division involves repeatedly dividing the first term of the polynomial by the first term of the divisor, then multiplying the result by the divisor and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor. Long division can be time-consuming and difficult for larger polynomials.
2. Factorization: Polynomial factorization involves factoring the divisor and the dividend, then dividing the two. This method can be useful for certain types of polynomials, but can also be time-consuming and difficult.
B. Advantages and disadvantages of synthetic division as compared to other methods
The main advantages of synthetic division over other methods of polynomial division are:
– It is faster and simpler than long division.
– It does not require factoring the polynomial or the divisor, as is the case with polynomial factorization.
The main disadvantage of synthetic division is that it can only be used to divide a polynomial by a linear factor of the form (x – a).
C. Examples to illustrate the differences between synthetic division and other methods
Here are some examples that illustrate the differences between synthetic division and other methods of polynomial division:
Example 1:
Divide 3x3 + 7x2 – 4x – 6 by (x – 1).
Synthetic division:
1 | 3 7 -4 -6
3 10 6
7 6 0
The quotient polynomial is 3x2 + 10x + 7, and the remainder is 0.
Long division:
3x2 + 10x + 7
x – 1 | 3x3 + 7x2 – 4x – 6
-3x3 + 3x2
————-
10x2 – 4x
10x2 – 10x
———-
6x – 6
6x – 6
—–
0
The quotient polynomial is 3x2 + 10x + 7, and the remainder is 0.
Polynomial factorization:
3x3 + 7x2 – 4x – 6 = (x – 1)(3x2 + 10x + 7)
The quotient polynomial is 3x2 + 10x + 7, and the remainder is 0.
Example 2:
Divide 2x4 + 6x3 – x2 – 13x – 10 by (x + 3).