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I NEED HELP ASAP....
The graph of a polynomial function of degree 10 has 5 x-intercepts, 4 of which have multiplicity 1 and one which has multiplicity 2. Describe the nature and number of all the function's zeros.
A) The function has 6 real zeros.
B) The function has 4 imaginary zeros.
C) The function has 6 real and 4 imaginary zeros.
D) The function has 5 real and 5 imaginary zeros.

Respuesta :

You're told there is a 10th degree polynomial. There are four roots of multiplicity 1 and one of multiplicity 2 (a double root).

Anytime there is a root of multiplicity 1 of a polynomial, its graph crosses the x-axis at that root. Anytime there is one of multiplicity 2, it means that we count the root twice and the graph has a tangency point.

The degree of a polynomial tells you how how many roots it has. Ours is degree 10, so it has ten roots. We have the four roots of multiplicity 1 and the one of multiplicity two, for a total of 6. (four and two).

So there are six real roots.

The rest of the roots are imaginary and non-real, and 10 - 6 = 4.  So there are four imaginary roots.


Thus choice C is best.

MrRoyal

The true statement is (c) The function has 6 real and 4 imaginary zeros.

The given parameters are:

[tex]\mathbf{Degree = 10}[/tex]

From the question, we have:

  • 4 intercepts with a multiplicity of 1
  • 1 intercept with a multiplicity of 2

The number of real roots is:

[tex]\mathbf{Real = \sum Intercept \times Multiplicity}[/tex]

So, we have:

[tex]\mathbf{Real = 4 \times 1 + 1 \times 2}[/tex]

[tex]\mathbf{Real= 4+ 2}[/tex]

[tex]\mathbf{Real= 6}[/tex]

The number of complex roots is:

[tex]\mathbf{Complex =Degree - Real}[/tex]

So, we have:

[tex]\mathbf{Complex =10 - 6}[/tex]

[tex]\mathbf{Complex= 4}[/tex]

Hence, the true statement is (c)

Read more about polynomial functions at:

https://brainly.com/question/11298461

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