I am familiar with this type of problem but can't seem to get the right answer.
Use the given zero to find the remaining zeros of each function
f(x)=x^4-9x^3+7x^2-91x-348,zero 5-2i
I normally would use synthetic division with the root 5-2i bringing it down to x^3. Then synthetic with 5+2i, bring it down to a quadratic solve the quadratic then I'll have zeros, but I can't get past the 5-2i with the synthetic division. Every time I do it I can't get it to equal zero. I have done it multiple times and it won't work.
One property of complex roots is that they always come in conjugate pairs.
So if one root is 5-2i, there will be another 5+2i
so there are two factors,
(x - 5 - 2i) and (x - 5 + 2i)
I multiplied these and go
(x^2 - 10x + 29)
Now do a long division of
(x^4-9x^3+7x^2-91x-348) by (x^2 - 10x + 29)
That should leave you with a quadratic, that can be solved for 2 more roots.
Let me know if it worked for you.
Finding the zeros of a polynomial function involves finding the values of x that make the function equal to zero. In this case, you are given one zero, which is 5-2i. To find the remaining zeros, you can follow these steps:
Step 1: Understand the complex conjugate property.
When a polynomial has complex zeros, they occur in conjugate pairs. For example, if a+bi is a zero, then a-bi is also a zero.
Step 2: Set up synthetic division.
Perform synthetic division using the given zero, 5-2i. The divisor becomes (x - (5-2i)).
Step 3: Find the conjugate of the given zero.
The conjugate of 5-2i is 5+2i. Thus, you will need to use synthetic division again, this time with the zero 5+2i.
Step 4: Solve the resulting quadratic equation.
After performing synthetic division with 5+2i, you will have a quadratic equation. Solve this equation to find the remaining zeros.
Step 5: Combine the zeros.
Finally, combine the zeros you found (including the given zero) to form the complete set of zeroes of the polynomial function.
If you are struggling with getting the synthetic division to equal zero with 5-2i, there might be an error in your calculations. Make sure you are correctly subtracting and multiplying during each step. Double-check your calculations and try performing the division again.
If you are still having trouble, please provide the intermediary steps and calculations you have made so far, so I can assist you further.