activity 8.counting the roots of polynomial equation

by inspection determine the number of real roots of each polynomial equation. Roots multiply n are counted n times

1.(x-4)(x+3)^2(x-1)^3=0
2.x^2(x^3-1)=0
3.x(x+3)(x-6)^2=0
4.3x(x^3-1)^2=0
5.(x^3-8)(x^4+1)=0

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Answers

Answer 1

Answer and Step-by-step explanation:

A root is considered a real root if it is not an imaginary number (a number in the form of a + bi, where i is a number that is not on the real number line, hence the name: imaginary). Let's inspect the given polynomials.

1. (x-4)(x+3)^2(x-1)^3=0, the equation implies

x – 4 = 0 or (x + 3)^2 = 0 or (x – 1)^3 = 0, this gives:

x = 4

x = –3 twice

x = 1 thrice

They are all real numbers, therefore it has 6 real roots.

2. x^2(x^3-1)=0

x² = 0

x³ – 1 = x³ – 1³ = (x – 1)(x² + x + 1) = 0

The quadratic expressions gives 2 imaginary roots. To check this we can use the determinant formula.

D = b² – 4ac = 1² – 4(1)(1) = 1 – 4 = –3 < 0. This gives roots that are not real.

Therefore the real roots of this polynomial are: 0(twice) and 1. Hence, it has 3 real roots.

3. x(x+3)(x-6)^2=0

x = 0

x = 3

x = 6 twice

Thus, it has 4 real roots.

4. 3x(x^3-1)^2=0

3x = 0 and this gives x = 0

(x^3 – 1)² = [(x – 1)(x² + x + 1)]² = (x – 1)²(x² + x + 1)² = 0, only two roots are real here (the quadratic expression has unreal roots, therefore 1 twice are the real roots).

Therefore, this has 3 real roots.

5. (x^3-8)(x^4+1)=0

x³ – 8 = x³ – 2³ = (x – 2)(x² + 2x + 4) = 0

The only real root here is 2, the quadratic expression had imaginary roots and can be checked using the determinant formula

D = b² – 4ac = 2² – 4(1)(4) = 4 – 16 = – 12 < 0.

x⁴ + 1 = 0 has imaginary roots. Therefore, this polynomial has only one real root.

Related Questions

Anna is buying school supplies. She spends $3 for a pack of 4 fancy pencils and $2 for a pencil case. Then, she buys a notebook for $2. She has a $10 bill. How much change will she receive?

Answers

1782375372927919171625828191 sry I need to answer a question

Answer:

3 not including tax, if it needs to include tax i dont know

help me please this do in 5 MIN

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Answer:

Step-by-step explanation:

Because 7/12 is more then a half

Also its obvious that 1/8 is the least used

(-7/10)=?/10 change to higher terms please help !!

Answers

Answer:

-7

Step-by-step explanation:

you cant change the numerator if you cant change the denominator

Which table represents a linear function?

Answers

Answer:

The top right option.

Step-by-step explanation:

Christian opened a savings account with $3250. His account will earn 5.2% simple interest.
How much money will be in his account after 6 years?

Answers

The total amount of money will be in his account after 6 years is $4264.

Given that, principal = $3250, rate of interest = 5.2% and time period = 6 years.

What is the simple interest?

Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time.

Simple interest is calculated with the following formula: S.I. = P × R × T, where P = Principal, R = Rate of Interest in % per annum, and T = Time, usually calculated as the number of years.

Substitute P=$3250, R=5.2% and T=6 years in simple interest formula

That is, S.I. = (3250 × 5.2 × 6)/100

= $1014

Amount = Principal + Simple interest

= 3250 + 1014

= $4264

Therefore, the total amount of money will be in his account after 6 years is $4264.

To learn more about the simple interest visit:

https://brainly.com/question/25845758.

#SPJ6

Help a random guy of the internet...that sounds weird

Answers

Answer:

I think 1 is yr answer

hope it helps.

please help brainlyest to correct

Answers

Answer:

[tex]22961.5 cm^{2}[/tex]

Step-by-step explanation:

[tex]V = \pi r^{2} h[/tex]

30/2 = 15 (to find radius, r)

[tex]V = 3.14 * 15^{2} * 325[/tex]

[tex]V = 22961.5 cm^{2}[/tex]

You are helping your friend research hybrid cars. You learn that on a long- distance trip, the Chevrolet Volt traveled 571.95 miles on 12.3 gallons of gas. You also learn that the Toyota Prius traveled 753.48 miles on 15.6 gallons of gas. Determine which car got better gas mileage.

A Toyota Prius

B Chevrolet Volt

C The Two Hybrids Are Tied for the best mileage

D Not enough information to determine The answer

Answers

Answer: A) Toyota Prius

========================================================

Explanation:

To get the mileage per gallon, you divide the number of miles over the number of gallons.

For the Chevy Volt, the mileage is 571.95/12.3 = 46.5 miles per gallon (mpg)

For the Prius, the mpg rating is 753.48/15.6 = 48.3

So the Toyota gets the better mileage per gallon.

---------

If you're curious why we divide the numbers like that, consider this example

You drive 200 miles using 5 gallons of gas. We can form the ratio

200 miles : 5 gallons

To get the mpg rating, we need to know how many miles we can drive on 1 gallon of gas. So we divide both sides by 5

200/5 miles : 5/5 gallons

40 miles : 1 gallon

This hypothetical situation leads to 40 mpg

What is the solution to the system of equations ?
Y=4x-10
Y=2

Answers

Answer:

x = 3

Step-by-step explanation:

If Y = 2, lets plug it in.

2 = 4x - 10

reverse operations,

4x = 12

x = 3

D is the answer.

Answer:

d (3, 2).

Step-by-step explanation:

Y=4x-10

Y=2

PLug y = 2 into the first equation:

2 = 4x - 10

12 = 4x

x = 3.

The solution (x, y) = (3, 2).

how do they know what numbers to pick

Answers

Answer:

casue i said so

Step-by-step explanation:

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Gereome has an hour and half before he has to leave to play basketball. If it takes him 7/12 an hour to do his homework and ⅗ of an hour to find his uniform, how much time will he have left?