TI-30Xa Algorithm: Synthetic Division
Introduction
Synthetic division is a fairly simple division algorithm that divides a polynomial by the nominal (x-a):
p(x) / (x - c) where
p(x) = a_n * x^n + a_n-1 * x^(n-1) + a_n-2 * x^(n - 2) + … + a_1 * x + a_0
c = a numerical constant
the result, q(x), is a polynomial of order n-1:
q(x) = b_n-1 * x^(n-1) + b_n-2 * x^(n - 2) + … + b_1 * x + b_0 + b_r / (x - c)
b_r = remainder term
where
b_n-1 = a_n
b_i-1 = b_i * c + a_i for i = n-2 down to r (“-1”, see the example below)
For example, for the polynomial:
p(x) = a3 * x^3 + a2 * x^2 + a1 * x + a0
q(x) = p(x) / (x - c)
p(x) has the order n = 3, so q(x) will have the order n = 3 - 1 = 2:
q(x) = b2 * x^2 + b1 * x + b0 + br / (x - c)
b2 = a3
b1 = b2 * c + a2
b0 = b1 * c + a1
br = b0 * c + a0 (the algorithm stops, this is the remainder term)
If br = 0, the x - c divides p(x) evenly, and x = c is a root of p(x).
Note: for any “missing” terms, fill the term with 0.
Example: x^3 + 4 * x - 5 becomes x^3 + 0 * x^2 + 4 * x - 5.
Calculator Algorithm
Store c in one of the memory registers 1, 2, or 3. We’ll call this memory register m for the purpose of the blog.
Note the first coefficient of q(x): a_n
Compute the rest of the coefficients as follows: [ × ] [ RCL ] m [ + ] a_ni [ = ]
Examples
Remember: m is memory register 1, 2, or 3, your choice.
Example 1: (21 * x^2 + 42 * x + 144) / (x - 12)
c = 12
a2 = 21
a1 = 42
a0 = 144
b1 = a2 = 21
12 [ STO ] m
Enter 21
b0:
[ × ] [ RCL ] m [ + ] 42 [ = ]
Result: 294
br:
[ × ] [ RCL ] m [ + ] 144 [ = ]
Result: 3672
Stop.
q(x) = 21 * x + 294 + 3672 / (x - 12)
Example 2: (2 * x^3 + x - 3) / (x - 3) = (2 * x^3 + 0 * x^2 + x - 3) / (x - 3)
c = 3
a3 = 2
a2 = 0
a1 = 1
a0 = -3
b2 = a3 = 2
3 [ STO ] m
Enter 2
b1:
[ × ] [ RCL ] m [ + ] 0 [ = ]
Result: 6
b0:
[ × ] [ RCL ] m [ + ] 1 [ = ]
Result: 19
br:
[ × ] [ RCL ] m [ + ] 3 [ +/- ] [ = ]
Result: 54
q(x) = 2 * x^2 + 6 * x + 19 + 54 / (x - 3)
Example 3: (4 * x^3 + 8 * x^2 - 5 * x + 3) / (x + 2)
c = -2
a3 = 4
a2 = 8
a1 = -5
a0 = 3
b2 = a3 = 4
2 [ +/- ] [ STO ] m
Enter 4
b2:
[ × ] [ RCL ] m [ + ] 8 [ = ]
Result: 0
b3:
[ × ] [ RCL ] m [ + ] 5 [+/-] [ = ]
Result: -5
br:
[ × ] [ RCL ] m [ + ] 3 [ = ]
Result: 13
q(x) = 4 * x^2 - 5 + 13 / (x + 2)
I think this is a fundamental algorithm for students and math enthusiasts to learn, and it’s fairly simple to get a hang of it.
Until next time,
Eddie
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