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Lana71 [14]
3 years ago
14

Let f be a functions of degree 4 whose coefficients are real numbers: two of its zeros are - 3 and 4 - i. Explain why one of the

remaining zeros must be a real number. Write down one of the missing zeros.
Mathematics
1 answer:
kvasek [131]3 years ago
8 0

Answer:

Step-by-step explanation:

We have the following theorem, if f is a polynomial with real coefficients, we can factor it completely in factors of the degree at most 2.

Consider first a polynomial of degree two, hence it is a polynomial of the form ax^2+bx+c. The cuadratic formula tells us that the solutions are of the form

x = \frac{-b\pm \sqrt[]{b^2-4ac}}{2a}.

Note that square root, over the reals, tells us that the are only real solutions if b^2-4ac \geq 0. If that is not the case, say it's negative, the solution are complex. Then, the solutions are of the form

x = \frac{-b \pm i \sqrt[]{4ac-b^2}}{2a}. NOte that this means that if we have a complex number of the form a+bi that is a solution, then the number a-bi (who is called the complex conjugate) is also a solution.

Recall that when we have a polynomial f(x) whose a zero is the number c, then we can factor f as follows f(x) = (x-c) * p(x) where p(x) is another polynomial of lesser degree .

So far, we know that -3 and 4-i are zeros of the function f. Note that we are missing two zeros. But, since complex numbers are zeros of polynomial only by pairs (that is the number and its conjugate are zeros), then, we must have that one of the missing 2 zeros is a real number. We have 4-i as a zero, then, its complex conjugate must be also a zero, i.e 4+i is a zero.

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Which answer best describes the complex zeros of the polynomial function?
marysya [2.9K]

<u>Answer-</u>

<em>D. The function has one real zero and two nonreal zeros. The graph of the function intersects the x-axis at exactly one location.</em>

<u>Solution-</u>

The given polynomial is,

f(x)=x^3-x^2+6x-6

The zeros of the polynomials are,

\Rightarrow f(x)=0

\Rightarrow x^3-x^2+6x-6=0

\Rightarrow x^2(x-1)+6(x-1)=0

\Rightarrow (x^2+6)(x-1)=0

\Rightarrow (x^2+6)=0,(x-1)=0

\Rightarrow x=\sqrt{-6},\ x=1

\Rightarrow x=1,\ \pm \sqrt6i

Therefore, this function has only one real zero i.e 1 and two nonreal zeros i.e ±√6i . The graph of the function intersects the x-axis at exactly one location i.e at x = 1

3 0
3 years ago
Let HH represent the number of hummingbirds and SS represent the number of sunbirds that must pollinate the colony so it can sur
Nikitich [7]

Answer: 7

Step-by-step explanation:

Here the given inequality that shows the number of humming birds(H) and the number of sunbirds(S)  that must pollinate the colony so it can survive until next year,

6 H + 4 S > 74

If This year, 8 hummingbirds pollinated the colony.

By putting H=8 in the above inequality,

We get,

6× 8 + 4 S > 74

⇒ 48 + 4 S > 74

⇒ 4 S > 26

⇒ S > 6.5

Hence the least number of sunbirds is 7.

7 0
3 years ago
Read 2 more answers
I dont know how to do this
Kipish [7]
So what you're going to do is just multiply how you were normal to play multiply the numerator times the other numerator then multiply the denominator's and that should get you your answer and then you can try and simplify it by finding if it can be divided equally then see if it is bigger or smaller
4 0
3 years ago
Write an expression for the apparent nth term (a(n)) of the sequence. (Assume that n begins with 1.)
expeople1 [14]

Answer:

34

Step-by-step explanation:

5 0
3 years ago
watermelon is shot out of an air cannon and travels 22 yards. How many feet does the watermelon travel? Which conversion factor
Firlakuza [10]

Answer:

it traveled 66 feet and a conversation factor u would use is 1 yard = 3 feet

Step-by-step explanation:

8 0
3 years ago
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