Find the complex zeros of the following polynomial function. Write f in the factored form.
F(x)=2x^4-5x^3-20x^2+115x-52
Look for the low-hanging fruit first. No easy roots, like 1 or 2, but we do find
F(x) = (x+4)(2x^3-13x^2+32x-13)
=(x+4)(2x-1)(x^2-6x+13)
Now use the quadratic formula for the complex roots.
This one was a little messy, since the roots took a little trial and error. Synthetic division comes in handy here.
To find the complex zeros of a polynomial function, we need to factor the polynomial.
First, we can apply the Rational Root Theorem to find any possible rational zeros. The rational root theorem states that if a rational number, p/q, is a zero of a polynomial function, then p is a factor of the constant term and q is a factor of the leading coefficient.
In this case, the leading coefficient is 2 and the constant term is -52. Possible rational zeros would be factors of 52 (±1, ±2, ±4, ±13, ±26, ±52) divided by factors of 2 (±1, ±2). So, the possible rational zeros are ±1, ±2, ±4, ±13, ±26, ±52.
Now we can use synthetic division or long division to check which of these possible roots are actual zeros of the polynomial.
Starting with f(x) = 2x^4 - 5x^3 - 20x^2 + 115x - 52, we perform synthetic division for each of the possible rational zeros until we find any actual zeros of the polynomial.
For example, let's try x = 1:
2 | 2 - 5 - 20 + 115 - 52
|_________________________
| 2 - 3 - 23 + 92 | 40
The remainder is 40, which means (x - 1) is not a factor of the polynomial.
We repeat this process for the other possible rational zeros until we find a zero or all possibilities have been tried.
After trying all possible rational zeros, we find that none of them are actual zeros of the polynomial. Therefore, this polynomial does not have any complex zeros.
To write the polynomial f(x) in factored form, we can use the Rational Root Theorem to determine the possible linear factors. Since there are no actual rational zeros, the factors would be of degree 2 and higher.
So, the factored form of f(x) would be: f(x) = (2x^2 + 7x - 4)(x^2 - 6x + 13)
Note: The quadratic factors can be expanded and factored further, but that is beyond the scope of finding the zeros of the polynomial.