Recently, I have seen one of my student, struggling with the Synthetic division method of polynomial division. I thought it would a good idea if I write the simplified process of the synthetic division in my blogs.
Additionally, I am happy to announce that I have started a student friendly Youtube channel (@SSMathTutoring) for Math lovers .. Hippie!!. 🙂 I will cover various math topics on this channel over the time which will help my students to learn those topics in their own time.
Synthetic division
Synthetic division is a simplified method used to divide polynomials. It is particularly useful when dividing a polynomial by a linear factor*, resulting in a polynomial of lower degree. This technique eliminates the need for long division and provides a more straightforward approach.
So, let’s dive in and explore the method :
What is Polynomial division ?
Polynomial division methods are techniques used to divide one polynomial by another. It is just same as dividing two numbers. such as 3/5 , or 1/2. Example of Polynomial divisions: (3x3+2x²-5)/(x-2) or (2x3– 5x²+ 3x – 1)/(x – 2)
How do we divide polynomials?
There are two methods commonly used to divide Polynomials : a) Long Division Method, b) Synthetic Division Method
To perform synthetic division, follow these steps :
Step 1 : Write the polynomial in descending order of powers, with any missing terms represented by zeros. For example : (x3– 2x²- 4x + 1)/(x – 1), i.e. Polynomial x3– 2x²- 4x + 1 is divided by Linear factor x – 1 . The coefficient of x3– 2x²- 4x + 1, are 1 , -2 , -4 and 1 in the descending order of powers.
Step 2: Identify the divisor, which should be in the form of “(x – a),” where “a” is a constant. Here, (X-2) is the divisor, x-1 = 0 -> x = 1
Step 3 : Set up the synthetic division table (as shown in the video), with the divisor’s constant term as the divisor and the polynomial’s coefficients as the dividend. Bring down the first coefficient to the bottom row of the table. Multiply the divisor by the value in the bottom row, and write the resulting product in the next row. Add the entries in each column and write the sum in the bottom row. Repeat this step until all coefficients have been used.
Step 4 : this is the answer step. The final row represents the coefficients of the quotient polynomial.
Synthetic division is an efficient method for dividing polynomials, especially when working with linear divisors. However, it can only be used to divide by linear factors and is not applicable for polynomials with complex or quadratic divisors.
*A linear factor is an algebraic expression of the form (x – a), where ‘a’ is a constant. In polynomial algebra, a linear factor represents a linear polynomial of degree 1. It is called a “factor” because when multiplied with other factors, it helps factorize or break down a polynomial into simpler terms. Linear factors play an important role in various mathematical concepts, such as polynomial division, finding roots of polynomials, and solving equations.
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