TI-BASIC:Synthetic Division - Revision history

Maintenance script: Automated superscript correction

2016-02-24T22:22:50Z

Automated superscript correction

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	Synthetic division works for problems in the following format.		Synthetic division works for problems in the following format.

	(a,,0,,x^^n^^ + a,,1,,x^^n-1^^ + a,,2,,x^^n-2^^ + ... + a,,n-1,,x^^1^^ + a,,n,,x^^0^^) / (x-r)		(a,,0,,x<sup>n</sup> + a,,1,,x<sup>n-1</sup> + a,,2,,x<sup>n-2</sup> + ... + a,,n-1,,x<sup>1</sup> + a,,n,,x<sup>0</sup>) / (x-r)

	Where a,,0,,≠0		Where a,,0,,≠0

	For example,		For example,
	Problem: find the roots of x^^3^^-x^^2^^-2x-12=0		Problem: find the roots of x<sup>3</sup>-x<sup>2</sup>-2x-12=0

	First we need to find all possible (r) in (x-r)		First we need to find all possible (r) in (x-r)
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	Input: 3→R		Input: 3→R
	Output {1,2,4,0		Output {1,2,4,0
	This means that (x^^3^^-x^^2^^-2x-12) / (x-3) = x^^2^^+2x+4		This means that (x<sup>3</sup>-x<sup>2</sup>-2x-12) / (x-3) = x<sup>2</sup>+2x+4
	(x-3) is a factor of (x^^3^^-x^^2^^-2x-12		(x-3) is a factor of (x<sup>3</sup>-x<sup>2</sup>-2x-12
	Then use the quadratic formula with x^^2^^+2x+4 to find the other factors.[[Category:TI-BASIC]]		Then use the quadratic formula with x<sup>2</sup>+2x+4 to find the other factors.[[Category:TI-BASIC]]
	[[Category:TIBD]]		[[Category:TIBD]]

KermMartian at 22:01, 24 February 2016

2016-02-24T22:01:18Z

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	[[TI-BASIC~~:Include inc~~:Routine\|picture=		{{Template:TI-BASIC:Routine\|picture=
	\|summary=This is useful for factoring large nth degree polynomials (and finding their roots, of course), as well as finding an upper and lower limit for the roots of an nth degree polynomial, as per the remainder and factor theorems and some others.		\|summary=This is useful for factoring large nth degree polynomials (and finding their roots, of course), as well as finding an upper and lower limit for the roots of an nth degree polynomial, as per the remainder and factor theorems and some others.
	It uses L₁ for input of the coefficients of the starting polynomial, uses X for r in (x-r) (what you're dividing by), so have those variables predefined (or put :Input L₁:Input X at the the beginning of the program), and it outputs L₂ as the answer with the last list entry as the remainder.		It uses L₁ for input of the coefficients of the starting polynomial, uses X for r in (x-r) (what you're dividing by), so have those variables predefined (or put :Input L₁:Input X at the the beginning of the program), and it outputs L₂ as the answer with the last list entry as the remainder.
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	\|author=TI-GBR, optimized by earthnite		\|author=TI-GBR, optimized by earthnite
	\|url=http://tibasicdev.wikidot.com/forum/t-1109444		\|url=http://tibasicdev.wikidot.com/forum/t-1109444
			}}

	~~[[code]]~~
	dim(L₁→dim( L₂		dim(L₁→dim( L₂
	L₁(1→L₂(1		L₁(1→L₂(1

Maintenance script: Automated @@ correction

2016-02-24T21:59:05Z

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	[[TI-BASIC:Include inc:Routine\|picture=@@		[[TI-BASIC:Include inc:Routine\|picture=
	\|summary=This is useful for factoring large nth degree polynomials (and finding their roots, of course), as well as finding an upper and lower limit for the roots of an nth degree polynomial, as per the remainder and factor theorems and some others.		\|summary=This is useful for factoring large nth degree polynomials (and finding their roots, of course), as well as finding an upper and lower limit for the roots of an nth degree polynomial, as per the remainder and factor theorems and some others.
	It uses L₁ for input of the coefficients of the starting polynomial, uses X for r in (x-r) (what you're dividing by), so have those variables predefined (or put :Input L₁:Input X at the the beginning of the program), and it outputs L₂ as the answer with the last list entry as the remainder.		It uses L₁ for input of the coefficients of the starting polynomial, uses X for r in (x-r) (what you're dividing by), so have those variables predefined (or put :Input L₁:Input X at the the beginning of the program), and it outputs L₂ as the answer with the last list entry as the remainder.

Maintenance script: Initial automated import

2016-02-24T18:06:05Z

Initial automated import

New page

[[TI-BASIC:Include inc:Routine|picture=@@
|summary=This is useful for factoring large nth degree polynomials (and finding their roots, of course), as well as finding an upper and lower limit for the roots of an nth degree polynomial, as per the remainder and factor theorems and some others.
It uses L₁ for input of the coefficients of the starting polynomial, uses X for r in (x-r) (what you're dividing by), so have those variables predefined (or put :Input L₁:Input X at the the beginning of the program), and it outputs L₂ as the answer with the last list entry as the remainder.
|input=''L₁'' - the coefficients of the polynomial you want divide by (x-r)
|input=''R'' - the value of r in (x-r)
|output=''L₂'' - the coefficients of the answer and the remainder is stored as the last list entry
|variable=L₁, L₂, A, R, Ans
|compatibility=TI-83/84/+/SE
|author=TI-GBR, optimized by earthnite
|url=http://tibasicdev.wikidot.com/forum/t-1109444

[[code]]
dim(L₁→dim( L₂
L₁(1→L₂(1
For(A,2,dim(L₁
L₁(A)+R(L₂(A-1→L₂(A
End
L₂

Synthetic division works for problems in the following format.

(a,,0,,x^^n^^ + a,,1,,x^^n-1^^ + a,,2,,x^^n-2^^ + ... + a,,n-1,,x^^1^^ + a,,n,,x^^0^^) / (x-r)

Where a,,0,,≠0

For example,
Problem: find the roots of x^^3^^-x^^2^^-2x-12=0

First we need to find all possible (r) in (x-r)
Factor a,,n,,:-12 {±1,±2,±3,±4,±6,±12
Factor a,,0,,:1 {±1
List of a,,n,,/a,,0,,:{±1,±2,±3,±4,±6,±12
this last list is the list of all possible r in (x-r)

Now for synthetic division.
See the [http://www.wikihow.com/how-to-divide-polynomials-using-synthetic-division wikihow website]

Using the program.

Input L1 with the polynomial's coefficients, {1,-1,-2,-12→L1
Input R for r in (x-r), 3→R

We get a L2 as an answer, the last entry in the list is the remainder, if this is 0, we know that (x-r) is a factor.The other entries in the list represent the coefficients of the quotient, a polynomial of 1 degree less than the starting polynomial.
so for starting polynomial f(x) divided by (x-r) , f(x)= (x-r) * q(x) +R, where q(x) is the quotient of 1 degree less than f(x) and R is the remainder. if R is =0 (x-r) is a factor of f(x) and f(x) can be broken down into q(x)*(x-r)

Now, remember we found all possible r's? Well, do synthetic division for each of those r's until the remainder (R, the last entry on L2) is 0. when you have that, you rewrite your equation, so in our problem 3 for r gives 0 for R so we would rewrite the problem, using the coefficients as:

Input: {1,-1,-2,-12→L1
Input: 3→R
Output {1,2,4,0
This means that (x^^3^^-x^^2^^-2x-12) / (x-3) = x^^2^^+2x+4
(x-3) is a factor of (x^^3^^-x^^2^^-2x-12
Then use the quadratic formula with x^^2^^+2x+4 to find the other factors.[[Category:TI-BASIC]]
[[Category:TIBD]]