Synthetic division is a powerful tool to solve polynomials, especially equations with a degree of four or higher. It is an alternative method that differs from the traditional method of factoring polynomials and looking for zeros. Synthetic division provides a smarter and easier way to solve polynomials. This article explains the basics of synthetic division and how it can be used to find zeros in polynomials.
What is Synthetic Division?
Synthetic division is an expression used to divide binomials. It is a quick and easy way to find the roots (zeros) of a polynomial equation instead of having to factor the equation into its roots. Synthetic division works for polynomials of a degree four or higher, and it uses the pattern of coefficients from the polynomial to form a line of numbers called the “divisor row”. The line ends in a remainder and/or a last number that can be used to determine if the equation has a zero – or if any real number would be a solution.
Steps in Synthetic Division
The first step in synthetic division is to write down the equation in descending order of exponent (from highest to lowest). Then, under the equation, set up a row that contains all the coefficients in their respective order and add a 0 at the end. This row will become the divisor row. Next, divide the first coefficient of the given equation by the first coefficient of the divisor row. Then multiply the resulting number by all of the coefficients in the divisor row and place in descending order below the divisor row. This is called the first dividend row.
Continue this process by either adding or subtracting (as indicated by the sign of the first coefficient) all of the coefficients from the first dividend row with the last coefficient from the previous dividend row and putting them in descending order below. This process repeats until a last coefficient of zero or one is reached, at which point synthetic division is done.
Applying Synthetic Division to Polynomials
After all the steps have been followed, synthetic division can be applied to polynomials of any degree by dividing the polynomial by its leading coefficient and then subtracting the highest exponent from all other exponents. After this has been done, the number of dividend rows that appear will equal the degree of the polynomial equation. In addition, examining the last number in each dividend row will let you know if there is a zero.
Finding the Zeros of a Polynomial
Using synthetic division, it is easy to find the zeros of higher-degree polynomials. To do this, you must look at the remainder in the last dividend row. If the remainder is 0, then you have found a zero. To find out which number corresponds to that zero, take all of the numbers (except for the last one) and divide them by the leading coefficient and then add that result to each exponent in descending order.
Examples of Polynomial Equations and Their Zeros
For example, consider a polynomial equation with degree four, such as 3x⁴ + 4x³ – is 8x² – 4x + 11. The steps with synthetic division to solve this equation are as follows:First, we write down the equation from highest exponent to lowest: 3x⁴ + 4x³ – is 8x² – 4x + 11.Then, set up divisor row: 3, 4, -8, -4, 0.Next, divide 3 by 3 for our first dividend row: 3, 4, -8, -4, 11.Then subtract all coefficients in dividend row from 11: 0, -7, 16, 8, 11.Finally subtract all coefficients in dividend row from 8: -3, 14, 8, 11.In this final dividend row, since the last number is not 0, this means there are no zeros.
How to Solve for Zeros with Complex Numbers
The zeros of a fourth degree polynomial are usually complex numbers with real and imaginary components. To find these zeros for a fourth degree polynomial equation whose divisor row ends in one (with no remainder), first find the sum of all coefficients in the polyonomial equation. Then divide this sum by four and add this result to each exponent in descending order until you have found out all four zeros.
Tips for Working with Polynomials and Synthetic Division
When working with polynomials and synthetic division there are few tips that could be useful: • Becoming familiar with synthetic division and its pattern will make working with polynomials much easier.• If you make a mistake while trying to solve a polynomial using synthetic division it will still work just as long as you redo it with accuracy. • Be very careful when dividing coefficients; any difference in signs needs careful attention as it could change a lot in the end result.• Synthetic division is not reliable for polynomials with a degree lower than four.
Potential Pitfalls with Synthetic Division
The main caveat when using synthetic division is that it is not appropriate for certain types of polynomials such as those that have more than four terms. In addition, rounding errors can lead to inaccurate answers and incorrect solutions. Finally, synthetic division requires attention to details such as signs and powers of coefficients; any miscalculation can lead to incorrect solutions.
Summary
Synthetic division is an effective tool to solve quickly polynomials with a degree four or higher. It’s an alternative method that differs from the traditional method of factoring polynomials and looking for zeros. The steps involved in using synthetic division are fairly easy to follow. Furthermore, you can use it to find zeros with complex numbers and to ensure accurate results there are some things you should take into account such as being very careful when dividing coefficients and making sure that all parameters of your equation match.
