Hello, good afternoon.
I need help with my precalc hw.
we are learning about finding zeros
I don't which method I should use for this problem:
if i is a zero of x^3+3x^2+ix+(4+i)
what do i do? synthetic divison? or long division?
when i did long division i got a werid polynomial with a lot of 'i's in them what do I do with that?
Thanks!
Well, looking at the i in the end, you know one root has to be i. Start with it, and do division, I might suggest synthetic division.
what do u mean looking atthe end "you know one root has to be i" what does that mean?
you get
x^2+(3+i)x+(4i-1)
what do i do next?
Hello! Good afternoon.
To find the zeros of a polynomial equation, synthetic division or long division can both be used. However, in this case, I would suggest using synthetic division since it is a more efficient method.
Here's how you can use synthetic division to find the zeros:
1. Rewrite the given equation, but with the zero (i) written out explicitly:
x^3 + 3x^2 + i*x + (4 + i) = 0
2. Set up the synthetic division table, keeping in mind that the coefficients of the equation are:
1, 3, i, 4 + i
| i | 1 | 3 | 4 + i |
___________________________
| | | | |
|________|_____|_____|________|
3. Since the degree of the polynomial is 3, we need to divide it by a polynomial of degree 1. In this case, we will use (x - i) as the divisor.
4. Write the conjugate of the divisor in a cell below the table:
x - i = 0
x = i
5. Perform the synthetic division by bringing down the first coefficient (1) in the leftmost cell:
| i | 1 | 3 | 4 + i |
___________________________
| | | | |
|________|_____|_____|________|
| | 1 | | |
6. Multiply the divisor (x - i) by the result in the bottom cell (1), and write it under the next coefficient (3):
| i | 1 | 3 | 4 + i |
___________________________
| | | | |
|________|_____|_____|________|
| | 1 | | |
| | | i | |
| |_____|_____| |
7. Perform the addition (summarized by +) between the coefficients:
| i | 1 | 3 | 4 + i |
___________________________
| | | | |
|________|_____|_____|________|
| | 1 | | |
| | | i | |
| |_____|_____| |
| | | 3 | |
8. Repeat steps 6 and 7 for each subsequent coefficient.
| i | 1 | 3 | 4 + i |
___________________________
| | | | |
|________|_____|_____|________|
| | 1 | | |
| | | i | |
| |_____|_____| |
| | | 3 | |
| | | |(3-i)(4+i)|
9. The last cell in the bottom row represents the remainder. If the remainder is zero, then i is a zero of the given polynomial.
In this case, since the remainder is not zero, i is not a zero of the polynomial x^3 + 3x^2 + ix + (4 + i).
By using synthetic division, you can check if a given value is a zero of a polynomial equation. Remember that zeros are the values of x that make the equation equal to zero.