Mathematics High School

Find all rational polynomials p(x) = x3 + ax2 + bx + c such that a, b, c are
roots of the equation p(x)=0

Answers

Answer 1

Answer:

p(x) = x³ +0x² +0x +0p(x) = x³ +x² -x -1p(x) = x³ -x² -2x

Step-by-step explanation:

If the roots of p(x) are a, b, c, then it factors as ...

  p(x) = (x -a)(x -b)(x -c)

and the expanded form is ...

  p(x) = x^3 -(a+b+c)x^2 +(ac +b(a+c))x -abc

The requirement that the coefficients match the roots means we have three equations in three unknowns:

  a = -(a+b+c)

  b = ac +b(a+c)

  c = -abc

The attachment shows the solutions to these equations. The solutions are found by making the substitutions ...

b = xc = ya = -(x+y)/2

This leaves us with two equations in two unknowns that can be graphed on a Cartesian plane. The graph of the second equation is a circle:

  x(1 -a -y) -ay = 0   ⇒   (x +1)^2 +y^2 = 1

The graph of the first equation is the union of the line y=0 and two hyperbolas. The points of intersection between this graph and the circle are ...

  (x, y) = (-2, 0), (-1, -1), (0, 0)   ⇒   (a, b, c) = (1, -2, 0), (1, -1, -1), (0, 0, 0)

The three polynomials that correspond to these values are shown above and in the attachment.

_____

Additional comment

There is an irrational fourth solution to the requirement that the roots match the coefficients.

Find All Rational Polynomials P(x) = X3 + Ax2 + Bx + C Such That A, B, C Are Roots Of The Equation P(x)=0

Related Questions

What is the distance from point D to point C?
A.1 unit

B.2 units

C.3 units

D.4 units​

Answers

Answer:

the distance from point D to point C.

B.2 units

Answer:

B:-2

Step-by-step explanation:

just count the box and you will get your answer

What is the name give to the angle pair 3 and 5?

Answers

Alternate Interior Angles
Alternative interior are congruent

Answer these 2 different questions ( i am so confused on this hellp!!)
6) All the people of a neighborhood pooled together and won the lottery. They won $10,000,000 and each person got a 0.02 share. How much money did each person receive?

7) Sally scored 9.007 in gymnastics. Jack scored 8.949. How much higher was Sally's score than Jack's?

Answers

6) above 7) 0.058 or 0.048

Mr. Rao's gas tank holds up to 1718 gallons of gas. It took 12.65 gallons of gas to fill his tank.

How many gallons of gas did Mr. Rao have in his tank before he filled it?

Enter your answer as a decimal or as a mixed number in simplest form in the box.

gal

Answers

4.475 or 4 19/40 gallons

How?

12.65=253/20

253/20= 12 13/20

////////

I need a similar denominator of 12 13/20 and 17 1/8.

12 13/20×2/1= 12 26/40

17 1/8×5= 17 5/40

17 5/40-12 26/40= 4 19/40

Therefore,

the answer is 4 18/40 gals

Answer:

4 19/20                      

Step-by-step explanation:

17 1/8 - 12 13/20 = 4 19/20

If I have a discriminate value of -3, how many solutions and what type will I have?

Answers

Answer:

0

Step-by-step explanation:

If b^2 - 4ac is less than 0, it will not have any real roots/solutions so you would not get any solutions in this case.

Write an inequality equation: Lyla has $100 in her bank account. She wants to buy presents that are $5 each. Wrapping paper will cost her $25. How many presents can she buy?

Answers

(100 - 25) / 5
15 presents

Can someone please help me with this one this is confusing to me :’) tysm!

Answers

Answer: 125 centimeters

1,250 millimeters

Step-by-step explanation:

'centi' means hundred. 100 centimeters go into a meter.

'milli' means thousand. 1,000 millimeters go into a meter. Easy math from there.

It’s 125 centimeters or 1,250 millimeters.

Some nuts cost $22/kg. How much would 150 g cost?

Answers

Answer:

$3.30

Step-by-step explanation:

1 kg = 1000 g

therefore, 150 g = 0.15 kg

multiply the cost by the amount

22 x 0.15 = 3.3, or $3.30

find the volume of this figure:

Answers

Answer:

= 5l. 66 in^3

Step-by-step explanation:

Area of cross section (A) = 1/2×b×h

=1/2 ×3.5×4.1

=7.175

Volume =AXH

= 7.175× 7.2

= 5l. 66 in^3

admission to the state fair costs $5 and each ride costs $0.75 if john and his three sons went to the fair and john spent a total of $56 how many rides did each person ride?

Answers

Answer:

t

Step-by-step explanation:

7.9 is what percent of 8? Round to the nearest tenth.

Answers

Answer:

98.75

Step-by-step explanation:

have a nice day

The answer is 100 percent.

What is the area of the figure?

Answers

Answer:

doc

Step-by-step explanation:

sdsc

Write inequalities for the following:
A) AJ makes at least 20 dollars

B) Lyla has no more than 70 barrettes

C) Rook takes at most 6 presents

D) River gives no less than 8 bones

Answers

Answer:

AJ. x<20

Lyla. x> or equal to70

Rook. x>or equal to 6

River. x<8

I hope that this helped

[tex] \tt\blue{Pwe \: \: \: POINTS }[/tex]​

Answers

Answer:

thank you pare XD

Step-by-step explanation:

(≧▽≦)(≧▽≦)

Costco is selling boxes of pomegranates for $18.99. if a box contains 6 pomegranates, what is the cost of one pomegranate?

Answers

Answer:

$113.94

18.99 multiplied by 6 is 113.94

Help ASAPPPPP ALGEBRA 2

Answers

This is cross multipication.  Multiply 117 times 36 together for that you would use a calculator. You get 4221. Next step is to multiply x and 12 together that would be 12x.

4221= 12x

Divide  by 12 on both sides:

x= 351

351 is # of trouts in the lake .

FINAL PART OF MY HW!!! PART 6!!! IT IS STILL UNIT RATE!!!!!​

Answers

Answer: Answer is on first screenshot

Step-by-step explanation:

Johnny deposited $8000 into his savings
account. How much interest is earned in
three years with an interest rate of 3%?

Answers

Answer:

$720

Step-by-step explanation:

I (Interest) = P (principal) x R (rate) x T (time, in years)

I = 8000 x .03 x 3

I = 720

Compare: The graphs below show the number of points you earn in each level of a game. Which games, if any, have a proportional relationship between the number of points you earn and the level of 7the game? In which game can you earn the most points in level 2? Explain your answer.

Answers

Answer:

Ok A and C are both proportional because it goes through the origin.

Step-by-step explanation:

If the line doesn't start or go through the origin will not be proportional. So If the line went through or started from the origin will be proportional. Which leaves B as Non-proportional graph.

Can someone please help me with this one? Thank you I appreciate it!

Answers

Answer:

25.92 sorry if i get this wrong!

Step-by-step explanation:

So to do the math properly you have to convert the 7.20 into a fraction as well so the 7$ will be a while number and the .20 would be over 100 being 20/100 since there’s 100cents in 1$, so that would be 7 20/100 and since the pounds is 3 3/5 the denominators have to be the same so you’ll have to divide the 20/100 until you have a same denominator so if you dive the numerator and denominator by 5 you get 4/20 (20/5=4, 100/5=20) then again you divide them both by 4 being 1/5 so that would be 7 1/5 x 3 1/5 so for each of them you multiply the bottom and add the top so 7x5 = 35+1= 36 so that’s your new numerator but the denominator remains the same so 36/5 and you do the same with 3 3/5 you multiply 3x5=15+ the 3 = 18/5 so then that would be 36/5 x 18/5 now you multiply both the numerators and denominators equaling 648/25 then that equals to 25.92$

Use the linear combination method to solve the system of equations. â€""3x 8y = 16 3x 4y = 8 What is the value of x? â€""1 0 1 2.

Answers

The value of x from the system of equation is 0

Given the systems of equation

–3x + 8y = 16

3x + 4y = 8

Using the elimination method.

Add both equations

-3x + 3x + 8y + 4y = 16 + 8

12y = 24

y = 24/12

y = 2

Substitute y = 2 into equation 2:

3x + 4y = 8

3x + 4(2) = 8

3x = 8 - 8

3x = 0

x = 0

Hence the value of x from the system of equation is 0.

Learn more on simultaneous equations here:https://brainly.com/question/185757

What is the difference between the simple and compound interest if you borrow $3,000 at a 6%
interest rate for 2 years?

Answers

Simple interest rate is based solely on the principal amount so total money owed = 3000*1.06
Compound is based on the principal amount and interest (interest on interest)
If we assume the money is compounded annually the total money owed would be
3000*1.06^2

Express 30^2/3 in simplest radical form.

Answers

Answer:

300

Step-by-step explanation:

Answer please................................

Answers

Answer:

1 and 4

Step-by-step explanation:

Nine out of ten students prefer math class over lunch. How many students do not prefer math if 200 students were asked

Answers

Answer:

i think probably 20

Step-by-step explanation:

if 200 students, students do not prefer math 10/100 × 200 = 20

The primary driver of offshore outsourcing is driven by __________.

Answers

Answer:

cost

Step-by-step explanation:

what is the greatest common factor of 32 and 56

Answers

its 8. hope this helps you.

13-n + 2n > 8 + 1 Solve the given inequality and graph and solution

Answers

because we are finding for the variable and not the coefficient we divide it by the number attached to the variable which is the coefficient to get the result of the variable

half the product of two number a and b, added to thrice a third number c

Answers

Answer:

  ab/2 +3c

Step-by-step explanation:

Maybe you want to translate this to an algebraic expression.

__

The product of a and b is written ab.

Half that can be 1/2ab or ab/2. The latter tends to be less ambiguous.

Thrice c is 3c.

So, your expression is ...

  ab/2 +3c

Express 29 and 37 as the sum of two squares.
If you can answer the correct answer first, then I will mark brainliest.

Answers

Answer:

See below

Step-by-step explanation:

[tex] {5}^{2} + {2}^{2} = 25 + 4 = 29 \\ \\ {6}^{2} + {1}^{2} = 36 + 1 = 37[/tex]

The values are 29 = 5² + 2² and 37 = 6² + 1².

What is square of number?

The definition of a square number states that they are numbers that have been multiplied by themselves. In other terms, a natural number is referred to as a square number if it is multiplied by itself.

Decimal squares are fairly simple to calculate, and the process is the same for both whole numbers (integers) and decimals.

Given numbers 29 and 37

to express in form of sum of two square

let = 29 = a² + b²

the perfect square number near 29 is 25

29 = 25 + 4

29 = 5² + 2²

and 37 = a² + b²

the perfect square near 37 is 36

37 = 36 + 1

37 = 6² + 1²

Hence the numbers in sum of two squares are,

29 = 5² + 2² and 37 = 6² + 1²

Learn more about square of number;

https://brainly.com/question/28610159

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