Mathematics, 22.10.2019 10:00 BrodsterBj

Trevor is studying a polynomial function f(x). three given roots of f(x) are –7, 2i, and 7. trevor concludes that f(x) must be a polynomial with degree 3. which statement is true?

trevor is correct.
trevor is not correct because –2i must also be a root.
trevor is not correct there cannot be an odd number of roots.
trevor is not correct because there cannot be both rational and complex roots.

Answers: 3

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Answer from: charrij26

For the polynomial wit hroots -7, 2i and 7 there are two possibilities:

the polynomial has real coefficients;the polynomial has complex coefficients.

Consider both these cases:

if the polynomial with real coefficients has complex root 2i, then the complex number -2i is also a root. Ttus, the polynomial of smallest degree will be of form: f(x)=a(x+7)(x-7)(x-2i)(x+2i)

. The degree of this polynomial is 4. As you can see in this case correct option is B.if the polynomial is with complex coefficients, then its smallest degree will be 3 and the form of polynomial will be: f(x)=a(x+7)(x-7)(x-2i)

. As you can see in this case correct option is A.

two possible options A and B.

Answer from: miguelsanchez1456

Option B

Trevor isn't correct because -2i must also be a root

Solution:

For the polynomial with roots -7, 2i and 7 their roots can be,

1. ) Real roots

2.) Imaginary roots

The real roots are: -7 and +7

The imaginary root given is: 2i

The imaginary roots come from the square root. So they will be in form of $\pm 2i$

Therefore,

For f(x) with roots -7 and +7 and $\pm 2$ we have,

f(x)=a(x+7)(x-7)(x-2i)(x+2i)

Fundamental Theorem of Algebra states that a polynomial will have exactly as many roots as its degree (the degree is the highest exponent of the polynomial).

So for f(x) with 4 roots, degree of f(x) is 4

So option B is correct. Trevor is not correct because –2i must also be a root.

Answer from: ijohnh14

The correct option is: Option (B) Trevor isn't correct because -2i must also be a root

Explanation:
The roots given are: -7, 2i, +7.

We need to differentiate between the real roots and the imaginary roots.

The real roots are: -7 and +7.
The imaginary root given is: 2i

However, the imaginary roots ALWAYS come in pair like $\pm 2i$
Because the imaginary roots come from the square root; In this case, the roots might be in the form of: $\sqrt{-4}$ . The square-root of the negative number yields the imaginary numbers (which can be positive or negative), which in this case is +2i and -2i.

Hence the correct answer is that Trevor isn't correct because -2i must also be a root (Option b).

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Trevor is studying a polynomial function f(x). three given roots of f(x) are –7, 2i, and 7. trevor c...