# Synthetic Division

• Jan 1st 2006, 12:40 PM
aussiekid90
Alright, i have to discuss how synthetic division can be used with any linear factor of the form ax+b. for example, any polynomial divided synthetically by ax+b. I didn't know that it was possible i thought you could only do x+b or x-b with no leading coefficient. How would you do it with any leading coefficiant. A probelm like: 4x^3-8x^2-11x+9 / 2x+3
• Jan 1st 2006, 02:25 PM
ThePerfectHacker
I do not know exactly what you are asking, but if you have a polynomial of degree $n>1$ then by synthetic division (assuming the remainder polynomial is zero) will have degree $n-1$. What you can do with this is the following useful trick.
Given a polynomial:
$f(x)=a_{0}+a_{1}x+...+a_{n}x^n$
divide it synthetically by $x-x_{0}$ where $x_{0}$ is a zero of the polynomial. Your polynomial must reduce to:
$g(x)=b_{0}+b_{1}x+...+b_{n-1}x^{n-1}$.
Now to find the zeros of $f(x)$ all need now is to find zeros of $g(x)$. This is especially useful when you are solving a cubic. Because the steps for solving a cubic a rather long and complicated if you can spot a trivial solution you can syntheticaly divide which will leave you with a quadradic equation. Now apply the quadratic formula and you have your 3 roots of the cubic.
• Jan 1st 2006, 02:38 PM
ticbol
Quote:

Originally Posted by aussiekid90
Alright, i have to discuss how synthetic division can be used with any linear factor of the form ax+b. for example, any polynomial divided synthetically by ax+b. I didn't know that it was possible i thought you could only do x+b or x-b with no leading coefficient. How would you do it with any leading coefficiant. A probelm like: 4x^3-8x^2-11x+9 / 2x+3

I have not studied before how to do synthetic division using (ax +b). It was by (x +a) only. So I'd reduce the (ax +b) into (x +c), then do it as usual.
In doing that, I'd divide all by the coefficient of the x in the (ax +b).

(4x^3 -8x^2 -11x +9) / (2x +3)
I'd divide everything by 2,
(2x^3 -4x^2 -5.5x +4.5) / (x +1.5)
and do the usual synthetic division.

..-1.5..|....2....-4....-5.5....4.5
.....................-3....10.5...-7.5
_____________________________
...............2....-7.....5.......-3 <----there is a remainder of (-3). That means (2x +3) is not a factor of 4x^3 -8x^2 -11x +9.
• Jan 1st 2006, 05:24 PM
ThePerfectHacker
Quote:

Originally Posted by ticbol
I have not studied before how to do synthetic division using (ax +b). It was by (x +a) only. So I'd reduce the (ax +b) into (x +c), then do it as usual.
In doing that, I'd divide all by the coefficient of the x in the (ax +b).

(4x^3 -8x^2 -11x +9) / (2x +3)
I'd divide everything by 2,
(2x^3 -4x^2 -5.5x +4.5) / (x +1.5)
and do the usual synthetic division.

..-1.5..|....2....-4....-5.5....4.5
.....................-3....10.5...-7.5
_____________________________
...............2....-7.....5.......-3 <----there is a remainder of (-3). That means (2x +3) is not a factor of 4x^3 -8x^2 -11x +9.

If you really want to you can express
$\frac{4x^3-8x^2-11x+9}{2x+3}$
as
$2x^2-7x+5-\frac{6}{2x+3}$
Thus your assertation that $2x+3$ does not divide $4x^3-8^2-11^x+9$ was correct. You reasoning by reducing $2x+3$ to $x+1.5$ then syntheticaly dividing lead to the correct solution that it does not divide it. I just divided it out without using your step. Meaning first divided $4x^3....$ by $2x...$ then multiplied result and subtracted... the entire synthetic algorithm but just with
$2x$
• Jan 1st 2006, 07:36 PM
ticbol
Quote:

Originally Posted by ThePerfectHacker
If you really want to you can express
$\frac{4x^3-8x^2-11x+9}{2x+3}$
as
$2x^2-7x+5-\frac{6}{2x+3}$
Thus your assertation that $2x+3$ does not divide $4x^3-8^2-11^x+9$ was correct. You reasoning by reducing $2x+3$ to $x+1.5$ then syntheticaly dividing lead to the correct solution that it does not divide it. I just divided it out without using your step. Meaning first divided $4x^3....$ by $2x...$ then multiplied result and subtracted... the entire synthetic algorithm but just with
$2x$

What synthetic algorithm? What you did was long division, not synthetic division.

Can you show how to do synthetic division using the (2x+3)? Let us see it.
• Jan 2nd 2006, 11:11 AM
ThePerfectHacker
There is a difference between synthetic divison and long division?
• Jan 2nd 2006, 05:33 PM
Jameson
Quote:

Originally Posted by ThePerfectHacker
There is a difference between synthetic divison and long division?

Yes. Synthetic division writes out the coefficients of the polynomial terms and uses a shortcut method to arrive at the quotient.