What is the remainder when 2x^3−9x^2+11 is divided by x−6?

Answer:First suppose that the roots of the equation

x2−bx+c=0(1)

are real and positive. From the quadratic formula, we see that the roots of (1) are of the form

b±b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2.

For the root or roots to be real, we require that b2−4c≥0, that is, b2≥4c. In order for them to be positive, we require that

b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√>0.

This immediately tells us that b>0, but we can go further. We can rearrange this to get

b>b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√,

which (assuming that b>0) is true if and only if

b2>b2−4c,

since both sides of the inequality are positive so we may square. But then

4c>0.

That is, if the roots are real and positive then b>0 and b2≥4c>0.

Now suppose that b>0 and b2≥4c>0.

Then the roots of (1) are real since b2−4c≥0, and b>0 guarantees that the root b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√ is positive.

So it remains to show that b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√>0. We have that

4c>0,

so that

b2>b2−4c,

then square rooting shows that

b>b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√,

so the roots of (1) are real and positive, as required.

Approach 2

the curve y equals x squared minus b x plus c showing two positive roots for y equals zero

This is intended to be a proof without words! We have from the diagram that:

If c, b and b2−4c are all positive, there are two real positive roots for x2−bx+c=0 (if b2=4c, we have two real positive equal roots).

If there are two real positive roots for x2−bx+c=0, then c and b are positive and b2−4c is non-negative.

(Why is the distance between the roots b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√?)

Approach 3

When solving problems about the roots of polynomials, it is often useful to find expressions those roots must satisfy and see if this tells us anything new. If α and β denote the roots of the equation, then

x2−bx+c=(x−α)(x−β)=x2−(α+β)x+αβ

and so α+β=b and αβ=c.

We also know that the roots of a quadratic equation are real if and only if the discriminant is non-negative, that is, if and only if b2−4c≥0.

Using these facts, if α and β are both real and positive, then b=α+β>0, c=αβ>0 and b2≥4c, as above.

Conversely, if b>0 and b2≥4c>0, then we know the discriminant is positive and hence both roots are real. We also have that

αβ>0(2)

and

α+β>0.(3)

As α and β are both real, by (2), we know that α and β are either both positive or both negative. However, if α and β were both negative, then (3) could not possibly hold. Hence α and β are both positive, as required.

We now sketch on a graph the region where b>0, c>0 and b2≥4c:

The curve b squared = 4 c as a quadratic with the c-axis vertical and the b-axis horizontal. The region below it is shaded.

The region of the b-c plane for which b>0, c>0 and b2≥4c

Sketch the region of the b-c plane in which the roots of the equation are real and less than 1 in magnitude.

We know that in order for the roots to be real we need b2≥4c as in the first part. We now need to find the region where

−1<b±b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2<1.(4)

We have b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2≤b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2 so we only need to consider the values for which both

−1<b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2andb+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2<1.

Firstly, we will consider the values for which

b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<2.

Rearranging gives

b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<2−b.

So b<2, as the square root is non-negative, and we can square both sides to get

b2−4c<4−4b+b2,

which we may rearrange to find c>b−1.

We will now consider the values for which

−2<b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√.

Similarly, we can rearrange to get

b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<b+2.

So b>−2, and we can, as before, square to get

b2−4c<b2+4b+4,

and hence c>−b−1.

To sketch the graph, we start by considering the boundary curves b2=4c, c=b−1 and c=−b−1, and the points at which they intersect. We can see that the two lines only intersect when b=0 and c=−1, and the lines intersect the curve when

b2=4b−4andb2=−4b−4

which rearrange to

(b−2)2=0and(b+2)2=0.

This tells us that each line intersects with the curve in only one place and so these lines must be tangent.

Is there a way we could have deduced this directly from (4)?

What do the lines being tangent signify in terms of our equation x2−bx+c=0?

Is this a representation of a well-known property of these equations?

Sketching the graph then yields the following picture:

The graph with the previous curve and the lines 4 c = 4 b minus 4 and 4 c = minus 4 b - 4. Each line touches the curve once and the two lines intersect at (0, minus 1). The region between the three lines/curves is shaded.

The shaded region is where b2≥4c, c>b−1, c>−b−1 and −2<b<2

We might notice that when we derived the inequalities c>b−1 and c>−b−1, a lot of the work we did was very similar.

We might also notice that the graph above is symmetric about the c axis. Why is this?

Does the graph give us any ideas about how we might deduce one inequality from the other?

Step-by-step explanation:

Complete question :

a researcher compared the Heights and shoe sizes for 50 men, selected at random. The equation shown describes a line of best fit for the data, where X is the shoe size and y is the height, in inches (y=1.6x + 58)

Based on the equation, what is the approximate shoe size for a man having a height of 60 inches

Answer:

1.25

Step-by-step explanation:

Given the regression equation :

y=1.6x + 58

X = shoe size ; y = height

Using the equation, the approximate shoe size of a man with a height of 60 will be ;

60 = 1.6x + 58

60 - 58 = 1.6x

2 = 1.6x

2/1.6 = 1.6x / 1.6

x = 2 / 1.6

x = 1.25

Hence, approximate shoe size of a man with a height of 60 will be 1.25

Answer:

- 2 [tex]x^{5}[/tex] + 7x²y - 24y²

Step-by-step explanation:

( 5 [tex]x^{5}[/tex] - 5x²y + 9y² ) - ( 7 [tex]x^{5}[/tex]  - 12x²y + 33y² ) =

5 [tex]x^{5}[/tex] - 5x²y + 9y² -  7 [tex]x^{5}[/tex]  + 12x²y - 33y² =

( 5 [tex]x^{5}[/tex] - 7 [tex]x^{5}[/tex] ) + ( - 5x²y + 12x²y ) + ( 9y² - 33y² ) =

- 2 [tex]x^{5}[/tex] + 7x²y - 24y²

Answer:

umm I think it 186

Step-by-step explanation:

Answer:

in six hours it would be $34.25

Step-by-step explanation:

The computation of the total cost for 6 hours is shown below:

For one hour it would be $8.50

For two hours

= $13.65 - $8.50

= $5.15

For three hours

= $18.80 - $13.65

= $5.15

For four hours

= $23.95 - $18.80

= $5.15

For five hours

= $23.95 + $5.15

= $29.10

For six hours

= $29.10 + $5.15

= $34.25

Hence, in six hours it would be $34.25

*answer was incorrect please see below*                                                                                                    

Answer:

BCE

Step-by-step explanation:

I hoped this helped and mark me as brainly pls

Answer:

a c b d

Step-by-step explanation:

Answer:

what kind of math?

Step-by-step explanation:

Answer:

It depends your grade, but I might be able to help.

Step-by-step explanation:

What type of math?

Answer:

u can use substitute or eliminate

Answer:

You can tell if a table is linear by looking at how X and Y change. If, as X increases by 1, Y increases by a constant rate, then a table is linear. You can find the constant rate by finding the first difference.

Step-by-step explanation:

Hope this helps!

Answer:

XOZ

OY

O

Step-by-step explanation:

just took it on edge

Answer:

There's 6 ways the boats can line up for the parade.

Step-by-step explanation:

Feel free to give brainliest.

Have an outstanding day!

Answer: 100. Can I get a Brainlest.

100 is the answer

Multiply.....

God Bless

Answer: $24.08

Step-by-step explanation: 25% off $35.00 is $22.50

and adding 7% tax which equals $1.50

so it adds up to 24.08

Answer:

For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

9514 1404 393

Answer:

Step-by-step explanation:

Each of the quantities is described in terms of the number of Schnauzers, and that is the number we are asked for. It is convenient to write an equation for the number of Schnauzers. We choose to use s for the variable representing that number.

  s = # of Schnauzers

  s+3 = # of Scotties, 3 more than the number of Schnauzers

  2s-5 = # of Wire-haired Terriers, 5 less than twice the number of Schnauzers

The total number of dogs is then ...

  (s+3) +(s) +(2s-5) = 78

  4s -2 = 78 . . . . simplify

  4s = 80 . . . . . . add 2

  s = 20 . . . . . . . divide by 4

There were 20 Schnauzers at the dog show.

Answer:

-2

Step-by-step explanation:

9g-4-5=-27

9g-9=-27

Add 9 to both sides

9g=-18

g=-2

Answer:

-18

if you need more info. just search up you question on Google and click the first link

Answer:

x is the hours worked and y is the money earned

Step-by-step explanation:

Usually the x variable is the independent variable

The independent variable is the variable that does not rely on anything

The y variable is the dependent variable

The dependent variable is the variable that relies on the independent variable

In this case the money earned depends on how many hours you work for

I hope this helps :)

Answer:

I believe that it would be x is hours worked and y is dollars earned but I could be wrong I skipped a year of math because I completed it so quickly now I cant remember some stuff

Answer:

-1,0

0,-2

slope is -2/1

Answer:

View the Screenshot that I have sent.

Answer:

58.76

Step-by-step explanation:

So, as you can tell 97,000 is the 100 percent of the people, in this case we don't know what 57,000 is. Just divide equation 1 by equation 2. Considering both have the same unit. I'm so sorry, if this doesn't explain the best way.

Answer:

x=5

Step-by-step explanation:

Answer:

Ishaan is 21. Christopher is 7.

Answer:  Ishann is 21 years old.

Step-by-step explanation:

In this situation, we will represent Ishaan age by the variable  y and we will represent Christopher's age by x.

Ishaan is 3 times as old as Christopher and that can be represented by the equation,   y = 3x  

He is also 14 years older, and it can also be represented by the equation,  

y = x + 14.  

y= 3x

y = x + 14  

Using both equations, substitute the value of y in one of the equations into the other.

x + 14 = 3x                   Solve for x .

-x           -x

 14 = 2x  

  x = 7  

This means that Christopher is 7 years old.

To solve for Ishaan's age , input the value of x into one of the equations.

y = 3(7)

y = 21.

Answer:

hi! i'm so sorry about your headache, hope it gets better !

still need help?

Step-by-step explanation:

Answer:

In this case, there are 11 letters in the word 'mathematics'; three letters appear twice: m, a, and t.

Step-by-step explanation:

I don't know why you want this but there you go

Answer: B: 4

Step-by-step explanation:

A number is divisible by 15 if the number ends on a 0 or a 5, and the sum of its digits is divisible by 3.

We want to find a 5 digit flippy number, this can be: modeled with:

N = ababa

Where a and b are single digit numbers.

If we impose that N must end with a zero, then we will have:

N  = 0b0b0 = b0b0

This is a 4-digit number, so we can discard all the options that end with a zero.

Then the only option that we have are the numbers like:

B = 5b5b5

This number will be only divisible by 15 if:

5 + b + 5 + b + 5 = K is divisible by 3.

Then let's try find the possible values of b.

K = 3*5 + 2*b

K has two terms, the left term is already divisible by 3, then K will be divisible by 3 only if the other term is also divisible by 3.

Then we want 2*b to be divisible by 3.

And 2 is a prime number, then b must be divisible by 3, and we know that b is a number between {0,1 , 2, 3, 4, 5, 6, 7, 8 ,9}

The options are:

2*0 = 0 is divisible by 3.

2*3  = 6 is divisible by 3.

2*6 = 12 is divisible by 3

2*9 = 12 is divisible by 3.

Then we have four values of b such that:

N = 5b5b5 is divisible by 15.

Then the correct option is: B: 4