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Sever21 [200]
3 years ago
13

What are the possible number of positive, negative, and complex zeros of f(x) = x6 – x5– x4 + 4x3 – 12x2 + 12 ?

Mathematics
2 answers:
FinnZ [79.3K]3 years ago
7 0
This is a polynomial with more than 2 as a degree. Using Descartes Rule of Signs: 
f(x) = x⁶ + x⁵ + x⁴ + 4x³ − 12x² + 12 
Signs: + + + + − + 2 sign changes ----> 2 or 0 positive roots 
f(−x) = (−x)⁶ + (−x)⁵ + (−x)⁴ + 4(−x)³ − 12(−x)² + 12 f(−x) = x⁶ − x⁵ + x⁴ − 4x³ − 12x² + 12 
Signs: + − + − − + 4 sign changes ----> 4 or 2 or 0 negative roots 
Complex roots = 0, 2, 4, or 6 
muminat3 years ago
7 0

Descarte's Rule of Sign is useful for finding the zeroes of a polynomial. The rule will tell you how many roots you can expect and of which type not where the polynomial's zeroes are. This rule is given as follows:

For \ a \ polynomial \ f(x)=a_{n}x^n+a_{n-1}x^{n-1}+ \ldots a_{2}x^2+a_{1}x+a_{0} \\ \\ with \ real \ coefficients \ and \ a_{0} \neq 0


\bullet \ The \ number \ of \ \mathbf{positive \ real \ zeros} \ of \ f \ is \ either \ equal \ to \\ the \ number \ of \ variations \ in \ sign \ of \ f(x) \ or \ less \ than \ that \\ number \ by \ an \ even \ integer. \\ \\ \bullet \ The \ number \ of \ \mathbf{negative \ real \ zeros} \ of \ f \ is \ either \ equal \ to \\ the \ number \ of \ variations \ in \ signs \ of \ f(x) \ or \ less \ than \ that \\ number  \ by \ an \ even \ integer.


That is, the function:

f(x)=x^6-x^5-x^4+4x^3-12x^2+12 \\ \\ \\ + \ - \ - \ + \ - \ + \\ \\ Has \ four \ changes \ in \ sign

4, 2, or 0 positive roots


On the other hand, the function:

f(-x)=(-x)^6-(-x)^5-(-x)^4+4(-x)^3-12(-x)^2+12 \\ \\ f(-x)=x^6+x^5-x^4-4x^3-12x^2+12 \\ \\ \\ + \ + \ - \ - \ - \ + \\ \\ Has \ 2 \ changes \ in \ sign

2, or 0 negative roots


Finally:

Complex roots: 0, 2, 4, or 6

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Need help on this problem
soldier1979 [14.2K]

Hey there!

Let's look at our graph here. It is a 4 x 8 chart, which means there are 32 total squares. 12 of these squares are highlighted in black. This gives us 12/32, which is equal to 3/8. Therefore, we can eliminate answer D as an option.

Now, we are looking for a percent of a fraction. Usually when you draw a chart that represents multiplying fractions, the bars will overlap. We do see 5 gray squares, but if that is all, this chart would not really represent anything. Let's say it goes all the way across the top, giving us 8 gray squares, 3 of them also being black.

This means we have 8/32, or 1/4 as our other fraction. As we said before, 3 of these squares are both black and gray. This is 3/32, so the answer to our problem has to be 1/12.

We have fractions 3/8 and 1/4.

\frac{3}{8}(\frac{1}{4})=\frac{3}{32}= \frac{1}{12}

Therefore, our answer is H.

I hope this helps!

3 0
3 years ago
What is the answer to the math equation p(7,9)
Harman [31]

Answer:

p > 63

Step-by-step explanation:

p/7-(9)>0

                p

Simplify   —

                 7

 p    

 — -  9  > 0

 7  

p - 63

 ——————  > 0

   7    

Multiply both sides by  7

Add  63  to both sides

            p > 63

4 0
3 years ago
Please answer quickly due in 5 MINUTES WILL MARK BRAINLIEST
AnnyKZ [126]
Crosssection= 1.7mx2=3.4
2x3=6x3=18

3.4+18=21.4

4 0
3 years ago
Solve for x <br> 5x=x+28 <br> simplify your answer as much as possible
Andrew [12]

Answer:

:.x=7 ans

Step-by-step explanation:

5x=x+28

5x-x=28

4x=28

x=28/4=7

8 0
2 years ago
Use the information from the polling agency to determine the sample size needed to construct a 95% confidence interval with a ma
Ber [7]

Answer:

the sample size n= 771

sample proportion is p = 1/2

Step-by-step explanation:

<u>Step1:</u>-

marginal error formula is \frac{2S.D}{\sqrt{n} }

but if 'n' is sample size then n = \frac{1}{(margin error)^{2} }

Given margin error is 36% = 0.036

n= \frac{1}{(0.036)^2}

on simplification , we get n= 777 polling agencies

<u>Step 2</u>:-

Margin of error = \frac{2\sqrt{p(1-p)} }{\sqrt{n} }

substitute values , we get

\frac{36}{100}  = \frac{2\sqrt{p(1-p)} }{\sqrt{771} }

after simplification, we get

\sqrt{p(1-p)} = \frac{1}{2}

squaring on both sides, we get

p(1-p) = \frac{1}{4}

multiply 'p' inside

p-p^{2} =\frac{1}{4}

cross multiplication '4' and simplify

4p^{2} -4p +1 =0 ........(1)

finding 'p' value

The equation (1) is of the form (a-b)^{2} = a^{2} -2ab+b^{2}

(2p)^{2} -2(2p)(1)+1^{2} = (2p-1)^2

(2p-1)^2 =0

p=\frac{1}{2}

<u>conclusion</u>:-

The sample size is 771

the sample proportion is p = \frac{1}{2}

4 0
3 years ago
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