given p(x) = x^3-2ix^2+4x-3i?

Question

use synthetic division to find p(3i)

Answer 1

no mames

Answer 2

I had never heard of the expression "synthetic division". Do you mean Ruffini's rule?

Divide the equation by (x - 3i) you get something like:
x^2 + ix + 1, with no remainder

In other words:
p(x) = x^3-2ix^2+4x-3i
becomes
(x^2 + ix + 1)(x - 3i) + 0 (0 being the remainder)

If x = 3i, then (x - 3i) = 0 and
(x-3i) multiplied by anything is still 0.

Therefore, the value p(3i) is only the remainder which, in this particular case, is 0.

Can we check?

p(3i) = (3i)^3 -2i(3i)^2 +4(3i) - 3i
p(3i) = -27i + 18i + 12i - 3i

seems to work

(but check my figures)


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jc:
it is a trick (although I know it under a different name) to solve complicated polynomials for a given value of x.

If you want to know the value of the polynomial (of whatever degree) for a given value x=a, then you divide the polynomial by (x-a) and you only need to solve for the remainder.

There are polynomials for which this method makes sense. In this case, the student is asked to practice it on a simple polynomial. Sure, it could be solved directly, but then the student would not learn about Ruffini's rule.

Answer 3

Use synthetic division by (x - 3i) to get a quotient q(x) and a remainder r. These will satisfy:
p(x) = q(x)*(x - 3i) + r
Evaluating this at x=3i gives:
p(3i) = q(x)*(3i-3i) + r = r
.
In other words, the remainder from dividing p(x) by (x - 3i) is equal to p(3i). You don't need the quotient from the division...just the remainder.

Answer 4

Wait, what? p(3i)=(3i)^3-2i(3i)^2+4(3i)-3i=-27i+18i+12i-3i

=0

What does synthetic division have anything to do with that?

Answer 5

uhh is this algebra 2?.. if it is ..i cud.. maybe.. solve it.. uhh.... guess and check :D lol

Answer 6

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