**Binomial Expansion**

Goal: To find the coefficients of when (ax + by)^n is expanded. The variables a and b are numeric.

(ax + by)^n = Σ (n nCr k *(ax)^(n-k) * (by)^k for k = 0 to n).

Example: (3x - 2y)^3 = 27 x^3 - 54 x^2 y + 36 x y^2 - 8 y^3

For the Casio Prizm and TI-84+, each coefficient is given in order. They are also stored in List 6. For the HP 39gii, a string is built representing the expanded binomial.

The list in the above example is {27, -54, 36, -8}

Casio Prizm:

POLYBINE

Binary Expansion

216 bytes

Lbl 6

"(AX+BY)^N"

"A"?→ A

"B"?→ B

"N"?→ N

If N≤0 Or Frac N≠0

Then

"N NOT AN INTEGER > 0" ◢

Goto 6

IfEnd

For 0 → K To N

ClrText

N nCr K × A ^ (N - K) × B ^ K → C

Locate 1,2,C

Locate 1,3,"×X^"

Locate 4,3,N-K

Locate 7,3,"×Y^"

Locate 10,3,K ◢

C→ List 1[K+1]

Next

Thanks to Ryan Maziarz for pointing out an extra quotation mark I had on line 3 (originally "A"?"→A). 2/6/2013

Note: The program will show the coefficients of the expansion one at a time. You can view the entire list of the coefficients in List 1.

TI-84+:

POLYBINE

Binomial Expansion

177 bytes

: Lbl 1

: Disp "(AX+BY)^N"

: Prompt A,B,N

: If N≤0 or fPart(N)≠0

: Then

: Disp "NEED INTEGER>0"

: Pause

: Goto 1

: End

: N+1->dim(L6)

: For(K,0,N)

: N nCr K*A^(N-K)*B^K->C

: ClrHome

: Output(2,1,C)

: Output(3,1,"*X^")

: Output(3,4,N-K)

: Output(3,7,"*Y^")

: Output(3,10,K)

: Pause

: C->L6(K+1)

: End

Note: The program will show the coefficients of the expansion one at a time. You can view the entire list of the coefficients in L6.

HP 39gii:

POLYBINE

11/23/2012

Polynomial Binomial Expansion

Expand (ax + by)^n

The results are returned in a string. Note, if N is not a positive integer, it is converted into one.

Input: POLYBINE(A,B,C)

EXPORT POLYBINE(A,B,C)

BEGIN

LOCAL S1,S2,K;

ABS(INT(N)) → N;

"" → S1;

FOR K FROM 0 TO N DO

string(COMB(N,K)*A^(N-K)*B^K) → S2;

S1 + "+" + S2 + "*X^" + string(N-K) + "*Y^" + string(K) → S1;

END;

dim(S1) → K;

right(S1,K-1) → S1;

RETURN S1;

END;

This blog is property of Edward Shore. 2012

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