Synthetic division is an algorithm used to quickly solve polynomial equations. This technique can be used to determine the quotient of the equation (X4 – 1) ÷ (X – 1), a fourth-degree polynomial that would normally take more time and effort to solve. In this article, we will explain step-by-step how to set up the equation and perform synthetic division, as well as how to interpret the results.
What is Synthetic Division?
Synthetic division is a type of short-hand algorithm used to quickly divide one polynomial by another. It enables a mathematician to divide larger polynomials without having to manually perform any long, tedious divisions. Instead, an equation can be divided using synthetic division in just four steps. This is also known as ‘long division without the hassle.’
Synthetic division is especially useful when dividing polynomials with a degree of three or higher. It is also useful for finding the roots of a polynomial, as the remainder of the division can be used to determine the roots. Additionally, synthetic division can be used to determine the factors of a polynomial, as the coefficients of the quotient can be used to determine the factors.
How to Set Up the Equation for Synthetic Division
To set up the equation for synthetic division, place the coefficients of the dividend (the expression being divided, i.e. X4 – 1) in descending order and then place a 0 at the end of the row. Below this row, write a vertical line followed by the coefficients of the divisor (X – 1).
Once the equation is set up, the synthetic division process can begin. The first step is to divide the first coefficient of the dividend by the first coefficient of the divisor. The result of this division is written directly below the dividend’s first coefficient. Then, multiply the divisor’s first coefficient by the result of the division and subtract this product from the dividend’s second coefficient. The result of this subtraction is written directly below the dividend’s second coefficient. This process is repeated until the last coefficient of the dividend is reached.
Calculating the Quotient Using Synthetic Division
Once you have the coefficients set up, you’re ready to begin synthetic division. The process is simple: you start in the second row and multiply the number in the first row by the number in the second row. Then place this number under the second row and add it to the third number in the first row. Place this number in the third row and repeat until you reach the end of the line.
Once you have completed the synthetic division, the last number in the bottom row is the quotient. The remaining numbers in the top row are the coefficients of the remainder. You can use these coefficients to calculate the remainder by substituting them into the original equation.
Understanding the Results of Synthetic Division
The bottom row of numbers will show you the remainder, while the quotient is represented by the numbers above it. Remember that your quotient should match the degree of the dividend (in this case, 4). If not, you’ve made an error.
In this example, we have a remainder of zero and our quotient is X3 + X2 + X + 1.
It is important to note that the remainder will always be a constant, regardless of the degree of the dividend. This means that if the remainder is zero, the equation will have a factor of (X – 0). If the remainder is not zero, the equation will have a factor of (X – remainder).
Other Uses for Synthetic Division
Synthetic division can also be used to find the zeroes (the values at which a polynomial is equal to zero) of a polynomial equation. To do this, divide the polynomial by (x – a), where ‘a’ is any number you choose. If the remainder is zero, then ‘a’ is a solution for that particular equation.
In addition, synthetic division can be used to divide polynomials of higher degree. This is done by breaking the polynomial into smaller polynomials of lower degree and then using synthetic division to divide each of them. This method is especially useful when the polynomial is too long to be divided using the traditional long division method.
Tips for Using Synthetic Division
Firstly, remember to always double check your work and look for any errors during setup and calculation as these can cause erroneous results. Additionally, be mindful of subtraction: sometimes when subtracting two numbers with similar values, you may end up with a negative number or a number greater than what you set out to subtract. Finally, remember that synthetic division only works for polynomials.
We hope this article has helped you understand how to use synthetic division to solve equations such as (X4 – 1) ÷ (X – 1), and what the calculated quotient is. With some practice, you’ll be able to quickly solve any polynomial equations using this technique!
