Write a mnemonic device that can help you remember the steps in synthetic division.?

Question

Synthetic division is a shortcut method of doing long division of polynomials when the divisor is of the form x+c. Long division with divisors of this for is the most commonly used kind because it is the long division that is needed to determine whether a number is the solution to an equation. This kind of long division is also useful for substituting in values to a polynomials, because of a theorem called the remainder theorem. The reason synthetic division works is that for this kind of division it is possible to greatly streamline down the process by cutting down on a lot of repetition, because of certain patterns that will always occur. To see how this works it is best to look at an example.
hope this helps becuase I surely dont know.. :/

Answer 1

Hi,

If synthetic division is to be fun,
Exponents inside must decrease by 1.
If there's a term that is not there,
Fill in, for example, a 0x square.
For the divisor remember to change the sign,
When that sign's switched, you'll be just fine.

Inside the problem it's not the worst,
Just look at the numbers and DROP the first.
Then MULTIPLY that times the number out front
and you'll be just fine,
if you WRITE your answer next column above the line.
And then it wouldn't be half bad,
If you'd take that column and just ADD.
Until you're done you won't be sad,
If you just MULTIPLY, WRITE, and ADD.

Suppose you wanted to divide 4x^3 - 6x - 5 by x - 3. You would put 3 out front for x - 3 (opposite sign) and coefficients inside, BUT this time there is no x^2 term. You must put in a zero in its spot to start the problem.


......__________
..3..| 4...0...-6...5
.......__________ DROP the 4 below the line.

.......__________
..3..| 4...0...-6...5
.......__________ MULTIPLY the number at the bottom times
the number out front.
WRITE the answer above the line in the next column.

.......__________
..3..| 4...0...-6...5
.......__12_______ ADD this column.
........4.....................

.......__________
..3..| 4...0...-6...5
.......__12_______
........4.12................
MULTIPLY the number at the bottom times the number out front.
WRITE the answer above the line in the next column.


.......__________
..3..| 4...0...-6...5
.......__12_36___ ADD this column.
........4.12................


.......__________
..3..| 4...0...-6...5
.......__12_36___
........4.12..30.........
MULTIPLY. the number at the bottom times the number out front.
WRITE answer above the line in the next column.

.......__________
..3..| 4...0...-6.....5
.......__12_36_90__ ADD this column.
........4.12..30........


.......__________
..3..| 4...0...-6.....5
.......__12_36_90_
........4.12..30...95... 95 is the remainder.

The coefficients of the quotient are 4, 12, and 30 which starts 1 power lower than the problem with an x^2. So the quotient is 4x^2 + 12x + 30 and the remainder is 95.

This can also be written as 4x^2 + 12x + 30 + 95/(x - 3)


So DROP, MULTIPLY, WRITE, and ADD, repeat the last 3 and you won't be sad.

I hope this helps! :-)

(I know it's not exactly a mnemonic device, but it was the best I could do.)

Answer 2

Could you please remind us of how to do synthetic division? It's been a long time.