All categories

3 years ago

Obtain the zeroes of polynomial

Physics

1 answer:

morpeh [17]3 years ago

3 0

I assume you meant to say

$f(x)=2x^4+3x^3-5x^2-9x-3$

Given that x = √3 and x = -√3 are roots of f(x), this means that both x - √3 and x + √3, and hence their product x ² - 3, divides f(x) exactly and leaves no remainder.

Carry out the division:

$\dfrac{2x^4+3x^3-5x^2-9x-3}{x^2-3} = 2x^2+3x+1$

To compute the quotient:

* 2x ⁴ = 2x ² • x ², and 2x ² (x ² - 3) = 2x ⁴ - 6x ²

Subtract this from the numerator to get a first remainder of

(2x ⁴ + 3x ³ - 5x ² - 9x - 3) - (2x ⁴ - 6x ²) = 3x ³ + x ² - 9x - 3

* 3x ³ = 3x • x ², and 3x (x ² - 3) = 3x ³ - 9x

Subtract this from the remainder to get a new remainder of

(3x ³ + x ² - 9x - 3) - (3x ³ - 9x) = x ² - 3

This last remainder is exactly divisible by x ² - 3, so we're left with 1. Putting everything together gives us the quotient,

2x ² + 3x + 1

Factoring this result is easy:

2x ² + 3x + 1 = (2x + 1) (x + 1)

which has roots at x = -1/2 and x = -1, and these re the remaining zeroes of f(x).

You might be interested in

A student pulls a block over a rough surface with a constant force FP that is at an angle θ above the horizontal, as shown above

gizmo_the_mogwai [7]

Answer:

B.The force of friction between the block and surface will decrease.

Explanation:

The force of friction is given by

$F_s = \mu N$

where $\mu$ is the coefficient of friction and $N$ is the normal force.

When the student pulls on the block with force $F_p$ at an angle $\theta$ , the normal force on the block becomes

$N = Mg- F_psin(\theta)$

and hence the frictional force becomes

$F_s = \mu (Mg- F_psin(\theta))$ .

Now, as we increase $\theta$ , $sin(\theta)$ increases which as a result decreases the normal force $Mg- F_psin(\theta)$ , which also means the frictional force decreases; Hence choice B stands true.

P.S: Choice D is tempting but incorrect since the weight $W=mg$ is independent of the external forces on the block.

6 0

3 years ago

Read 2 more answers

The gravitational force between two objects has a magnitude of F. If both masses were doubled and the distance between them doub

Fofino [41]

Answer:

F' = F

Explanation:

The gravitational force of attraction between two objects can be given by Newton's Gravitational Law as follows:

$F = \frac{Gm_1m_2}{r^2}$

where,

F = Force of attraction

G = Universal gravitational costant

m₁ = mass of first object

m₂ = mass of second object

r = distance between objects

Now, if the masses and the distance between them is doubled:

$F' = \frac{G(2m_1)(2m_2)}{(2r)^2}\\\\F' = \frac{Gm_1m_2}{r^2}$

F' = F

7 0

2 years ago

Can atoms go bad?, not in the reversible way like ionization and isotopes but really malfunction our die.

Elza [17]

Answer:

atoms cannot go bad

Explanation:

Because they stay alive and get good nutriants

5 0

3 years ago

8. At what position does the mass have the greatest acceleration?

gulaghasi [49]

Answer:

Option (e)

Explanation:

If a mass attached to a spring is stretched and released, it follows a simple harmonic motion.

In simple harmonic motion, velocity of the mass will be maximum, kinetic energy is maximum and acceleration is 0 at equilibrium position (at 0 position).

At position +A, mass will have the minimum kinetic energy, zero velocity and maximum acceleration.

Therefore, Option (e) will be the answer.

6 0

2 years ago

The six statements below represent Newton's three laws of motion and Kepler's three laws of planetary motion. Match each stateme

mote1985 [20]

Answer:

1. Force = mass x acceleration - Newton

2. A planet moves faster in the part of its orbit nearer the Sun and slower when farther from the Sun, sweeping out equal areas in equal times - Kepler

3. For any force, there is an equal and opposite reaction force - Newton .

4. An object moves at constant velocity if there is no net force acting upon it - Newton

5. The orbit of each planet about the Sun is an ellipse with the Sun at one focus - Kepler.

6. More distant planets orbit the Sun at slower average speeds, obeying the precise mathematical relationship p2-a3 - Kepler.

Explanation:

The three laws of planetary motion formulated by Johannes Kepler or Kepler's laws of planetary motion:

The first law states that the planets move around the Sun in an elliptical orbit with the Sun at one of the foci.
The second law states that the line segment joining a planet to the Sun sweeps out equal areas in equal time.
The third law states that the square of the orbital period (p) of a planet is directly proportional to the cube of the mean distance (a) from the Sun (or semi-major axis of its orbit) i.e., p² is proportional to a³.

The three laws of motion formulated by Sir Isaac Newton or Newton's laws of motion:

The first law, also known as the law of inertia states that an object at rest or moves at a constant velocity will remain at rest or keep moving at a constant velocity unless it is acted upon by a force.
The second law states that the total force (F) applied on an object is directly related to the acceleration (a) of that object produced by the applied force and the mass (m) of the object, i.e., F = ma (assuming the mass m is constant).
The third law, also known as the law of action and reaction states that when an object exerts a force on another object, then the latter exerts a force equal in magnitude and opposite in direction on the former object i.e., for every action, there is an equal and opposite reaction. The example includes the recoiling of a gun when it fires a bullet forward.

5 0

3 years ago