f(x)=5x^3-14x^2-19x-20

Question

Fundamental theorem of Algebra

Answer 1

Several of the rational zeroes are factors of 20: 1, 20, 4, 5.

Other rational zeroes are ⅕, ⅘, because the factors of the leading coefficient 5 are 1 and 5.

There are a further six rational zeroes formed by negating the six positive ones.

When we plug each of those into f(x) we discover f(4)=0, making x=4 a true zero.

Now divide by this zero:

4 | 5 -14 -19  -20

     5  20  24 | 20

     5    6    5 |  0  = 5x²+6x+5.

To find the other (complex) zeroes use the quadratic formula:

x=(-6±√(36-100))/10=(-6±√-64)/10=-⅗±⅘i.

The zeroes are: 4, -⅗+⅘i, -⅗-⅘i.