A new discovery
Synthetic division is a method used to divide any polynomial by a monic linear polynomial. Now, what does monic mean? Well, monic basically means that the coefficient of the maximum power of x, is 1. So, for those of you who do not know what synthetic division is, here is how it works:
As an example, let us take the divisor as x-2 and the dividend as 4x⁴-2x³-3x²+2x-1. First you have to put the value of x that makes the divisor equal to 0, here 2, on the left hand side of the grid.
Then, we must place the coefficients of all the terms of the dividend: 4, 2, -3, 2 and -1, on the right hand side of the grid.
Next, we should add 0 to the first coefficient, 4, and write the sum below. Then, we should multiply this sum to 2, place it under the next number and add again. This process should be repeated till the last coefficient, -1.
Now, the quotient of this division, 4x⁴-2x³-3x²+2x-1 by x-2, is 4x³+6x²+9x+20. And, the remainder is 39. Synthetic division can be used for a quadratic divisor as well, but the method seemed a little complicated to me and it didn’t work every time. So, I developed an easier method (in my opinion) to use synthetic division for a monic factorizable quadratic divisor. Here it is:
Note: This method only works if the quadratic is factorizable. Let us take the polynomial as 2x⁵+5x⁴-2x³-3x²+2x-1, and the divisor as x²+2x-15. First, we should factorize the divisor. It factorizes to (x-3)(x+5).
Next, we have to draw up a grid like the previous one, just 2 grids on top of each other. Then, we should put the coefficients of the dividend on the right hand side, and the roots one after the other vertically on the left hand side in any order.
Then we perform the same steps as before, only twice.
Now, the quotient is 2x³+x²+26x-40. And the remainder is 472x+z. To calculate z, we need to use the following rules: Let the number in the bottom right corner, here -1545, be m. And, let the number to its left, here 472, be l. The rules to calculate z are: z = m-yl And to calculate y, we just have to add the two roots a and b! y = a+b Completing the above example, the quotient and remainder are:
Using the formula P(x) = Q(x)D(x) + R(x), we will now verify the answer:
Let us see a few more examples to make it more clear (we will use the same dividend for simplicity): Both roots positive:
The verification:
Both roots negative:
The verification:
One root positive, the other negative and the greater magnitude root is positive:
The verification:
This shows that my method works! The rules aren't that hard to remember so have fun using it in class next time!