Thanks so much your method worked perfectly!

My solution was: the remaining zeros were 5+2i, 3, and -4

I have one question that's bugging me though, why is it that synthetic division did not work for me? I thought that long division and synthetic division were both interchangeable.

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.

Pre Calc - Reiny, Friday, November 13, 2009 at 11:21pm
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.

I'm glad to hear that my previous explanation helped you find some of the zeros of the function!

To address your question about synthetic division and long division, they are indeed both methods of dividing polynomials, but they have some differences in terms of their application and purpose.

Long division is a more general and widely applicable method of polynomial division. It involves the step-by-step division process that you may be familiar with from elementary school. In the case of finding zeros of a polynomial, long division can be used to divide the given polynomial by a linear factor to obtain a quotient polynomial. This quotient polynomial can then be further factored to find the remaining zeros.

On the other hand, synthetic division is a simplified form of polynomial division that can only be used when dividing by a linear factor of the form (x - c), where c is a constant. Synthetic division is often used when the divisor is a linear factor, as it is a quicker and more efficient method than long division in these cases.

In your case, you attempted to use synthetic division with the root 5-2i, but synthetic division only works with real numbers. This is because the coefficients in the synthetic division process are added or subtracted, and the complex numbers do not have a natural addition or subtraction operation. Therefore, synthetic division cannot be directly used with complex numbers as divisors.

To overcome this limitation and find the remaining zeros, you can use the fact that complex roots always come in conjugate pairs. This means that if 5-2i is a root of the polynomial, then 5+2i must also be a root. By multiplying the two conjugate roots together, you can obtain a quadratic factor.

Once you have this quadratic factor, you can then use either synthetic division or long division to divide the original polynomial by the quadratic factor to obtain a remainder that is a quadratic polynomial. Solving this quadratic polynomial will give you the remaining zeros of the function.

I hope this helps clarify the differences between synthetic division and long division, and why synthetic division did not work in your case. If you have any further questions, please let me know!

Synthetic division and long division are both methods used to divide polynomials. While they serve the same purpose, there are some differences between them.

Synthetic division is a shortcut method that is used specifically when dividing by a linear factor of the form (x - a), where "a" is a root or zero of the polynomial. It involves writing the coefficients of the polynomial in a specific way and performing a sequence of calculations to determine the quotient and remainder.

Long division, on the other hand, is a more general method that can be used to divide any two polynomials. It involves a series of steps similar to those used in long division with numbers. Long division can be used to divide by any factor, regardless of its form.

In the case of your problem, synthetic division did not work because the root you were trying to divide by was not a linear factor. The root 5-2i is a complex number, and complex roots always come in conjugate pairs. This means that if one root is a+bi, the other root is a-bi. In your case, the conjugate pair of 5-2i is 5+2i.

To find the remaining zeros, you need to factorize the polynomial using the linear factors corresponding to the given root and its conjugate. In this case, the factors are (x - 5 - 2i) and (x - 5 + 2i). Multiplying these factors together, you get (x^2 - 10x + 29).

Now, you can perform long division of the original polynomial by (x^2 - 10x + 29). This will give you a quadratic equation, which you can solve for the remaining zeros.

By using long division, you should be able to find the remaining zeros and solve the problem.