Last Updated on September 23, 2026 by Maged kamel
What is synthetic division? Numerical Analysis-part 2
As an introduction to numerical analysis, we will review synthetic division. Synthetic division is a shorthand, or shortcut, method of polynomial division. In the special case of dividing by a linear factor, it only works then.
Synthetic division is generally used not to divide factors but to find zeroes (or roots) of polynomials.
In synthetic division, to find the zero of the function, we sometimes guess the Value of x using so-called rational roots. The Rational root theorem Theorem is a handy way to generate a List of initial guesses when you are trying to find the Zeros of a polynomial.
A solved problem #1: synthetic division.
In our example. Y = x^2 – 5x + 6. We have a leading coefficient of x^2 = +1. This one can be analysed at the multiplication (+1+1)=1, and( -1-1 )will give us 1, while the 6, the constant, can be analysed as either (1*6) or(-1*-6) or (+2*+3) or (-2*-3).
So we have 1, and 6 + and – will go here.
And for the leading coefficient, we have 1 and 1 + and -(1*1),(-1*-1).
We put it down here in the denominator and a comma; plus or minus, we have 3 either plus or minus divided by the leading coefficient. Now we will proceed with synthetic division. We write onlythe coefficients, ,without x^2 or x. The coefficient of x^2 is 1.
We put (+1*+6),(-1*-6),(+2*3),(-2*-3).The first guess is (x+3)=0. X will be = -3;; it itwill go there. We put -3 on the left side. The coefficient of x^2, or the first coefficient, will go down as 1.
Then we multiply one by -3; we put it here, add -5 to -3, and get -8. Then we multiply -3 by -8. -8 multiplied by -3 gives +24, which we put here, and we add 24+6. We get +30; as long as we don’t get zero, this point is not a root, so we proceed to guess another Value. So these are the steps.
1-Our next guess is (x-2)=0; the coefficient of x ^2 will be carried down, multiplied by (+2), then add -+5 to add to +2, and we will get -3.
2-(-3) to be carried down, again multiply by(+2), we will have (-6), add to +6, we get a zero, this means that (x-2) is a zero point. Our solution x- 2 is one of the solutions.
3- We utilize (x-3). The next guess is (x-3). We use +3; 1 will go down. Multiply (+13)=3. Add to (-3); we get zero Value.
4- the two roots are (x-2) and (x -3 ). Now multiply them by each other, and you will have (x^2-5x+6) as the same function y=(x^2-5*x+6).
The next slide image shows the graphical solution of the previous function, which tabulates different values of x and estimates the corresponding y Value.
A solved problem for a synthetic division with a remainder.
The next slide image shows the division of f(x)=x^3-2x^2+x-3 by(x-2) using synthetic division. The remainder will be (-1).
A solved problem for dividing a given function by a Value.
Our new function is
1- It is required to divide the function by(x+2).
2- We have the following coefficients: for x^5 is 4, for x^4=0, for x^3=+6, for x^2=-9, for the constant is +6
3- We start with the solution of (x+2)=0; we divide by (-2).
4- Number 5 will continue down, multiplied by -2; the Product, which is -10, will be placed as a second item and added to +6; the Value will be -4.
5- (-4) is to be multiplied by -2, and the Product, which is=+8, will be placed as a third item.
6- Add +8to(-9), and the Value is -1 to be placed down.
We proceed to step 8, as shown in the next slide image.
Since we have no zero Value, we express the remainder as shown in the next slide image.
A solved problem for dividing a given function using long division.
We will solve the same problem using the long division procedure; instead of putting (-2), we will use (x+2).
Write 5x^4+6x^3-9x^2-7x+6 on the right side of (x+2). Divide 5x^4/x=5x^3.
Multiply(5x^3)*(x+2)=5x^4+10x^3. Subtract this Value from (5x^4 + 6x^3). The result will be -4x^3.
Divide -4x^3/x=-4x^2.
Multiply(-4x^2)*(x+2)=-4x^3-8x^2.+10x^3. Subtract this Value from (- 4x^3- 9x^2). The result will be -x^2.
Divide -x^2/x=-x.
Multiply(-x)*(x+2)=-x^2-2x. Subtract this Value from (-x^2-7x+6. The result will be -5x+6.
Divide -5x /x=-5. Multiply( -5 )*(x+2)=-5x-10.
Subtract (- 5x-10) from (- 5x+6). The result will be 16. There will be a remainder of 16/x+2. The full details are shown in the next slide image.
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The following Post concerns the Bisection method, a numerical method for root finding.
A very important source is Holistic Numerical Methods.