How to Do Synthetic Division in Plain English
Synthetic division is a mathematical technique used to divide polynomials. It is a simpler and faster method than long division, which makes it a popular method in algebra. In this article, we will discuss how to do synthetic division in plain English.
What You Need to Know Before Starting
Before we dive into the steps for synthetic division, you need to know the following terms: Polynomial: A mathematical expression consisting of variables and coefficients, which can be combined using addition, subtraction, multiplication, and division. Degree: The highest power of the variable in a polynomial. For example, in the polynomial 4x^3 + 2x^2  5x + 1, the degree is 3. Divisor: The polynomial used to divide another polynomial. Remainder: The leftover value after the polynomial division.Steps to Perform Synthetic Division
Follow these steps to perform synthetic division: Step 1: Write the polynomial in standard form. For example, let's say we have to divide 2x^3  5x^2 + 3x  7 by x  2. We will write the polynomial in the standard form as follows:
2  2 5 3 7
0 4 2 10
Note: The divisor x  2 is written as 2 because the opposite of 2 is used in the calculation. Step 2: Write the coefficients of the polynomial in the first row of the synthetic division table. The first row should contain the coefficients of the polynomial in the order of decreasing degrees. For example, in our case, the first row will be:
 2 5 3 7
Step 3: Write the root of the divisor in the second row of the synthetic division table.
The second row should contain the root of the divisor. In our case, the root is 2, so the second row will be:
2  2 5 3 7

2
Step 4: Multiply the root with the first coefficient of the polynomial and write the result below the first coefficient.
In our case, we will multiply 2 with 2 and write the result below 5, which is 4:
 2 5 3 7

2
4
Step 5: Add the result to the second coefficient of the polynomial and write the result below the second coefficient.
In our case, we will add 4 to 5 and write the result below 3, which is 1:
 2 5 3 7

2
4 1
Step 6: Multiply the root with the previous result and repeat the process until you reach the last coefficient.
In our case, we will multiply 2 with 1 and write the result below 7, which is 14:
 2 5 3 7

2
4 1 14
Step 7: Write the final result in the last row of the synthetic division table.
The last row will contain the coefficients of the quotient and the remainder. In our case, the last row will be:
 2 5 3 7

2
4 1 14
2x^2  x  7 + (14)/(x2)
Note: The quotient is 2x^2  x  7, and the remainder is 14/(x2).
FAQs
Here are some frequently asked questions about synthetic division:Question  Answer 

Can I use synthetic division for all polynomial divisions?  No, synthetic division can only be used when dividing by a linear polynomial of the form xa. 
Can I use synthetic division if the divisor has a degree greater than 1?  No, synthetic division only works when dividing by a linear polynomial. 
What is the advantage of using synthetic division over long division?  Synthetic division is a simpler and faster method than long division, which makes it a popular method in algebra. 
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