1. how many complex roots does 2x^4 + 3x^3 - x^2 + 5 = 0 have?

2
3
4
5

2. use the function to answer the question.
f(x) = x^3 + 5x^2 + 7x + 3 = (x + 1)(x + 1)(x + 3)

what is the multiplicity of the root 1?

1
2
3
4

4. how many real roots does x4 - x3 - 9x2 + 7x + 14 = 0 have?

1
2
3
4

1) Answer 4.

Explanation.

The number of total roots of a polynomial is the degree of the polynomial. In this case it is 4.
 
The real, i.e. non-complex roots of a polynomial correspond to the  x-intercepts of the graph. Then you can graph the polyomial to see how many real roots it has.

The graph of this polynomial shows that it does not cross or touch the x-axys which means that there are not real roots. Thereafter the 4 roots are complex.

2) The multiplicity of a root is the number of times that the corresponding factor appears in the polynomial. In this case the factor x + 1 appears two times, then - 1 has multiplicity 2.

3) You can either graph the polynomial or factor it.

If you factor it you will find:

x^4 - x^3 - 9x^2+7x+14 = (x +1)(x -2)(x^2 -7).

The there are 4 zeroes or 4 real roots.

 

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