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3 years ago

Which of the following represents the set of possible rational roots for the polynomial shown below 2x^3+5x^2-8x-20=0

Mathematics

2 answers:

kvasek [131]3 years ago

8 0

Answer:

± 1/2,±1, ±2,± 5/2, ±4 ,±5 , ±10, ± 20

Step-by-step explanation:

We can use the rational root theorem to find all the possible roots

2x^3+5x^2-8x-20=0

Let the constant term be called p and the leading term be called q. Then the possible roots are the positive and negative roots of the factors of p/q

p = 20

q = 2

Factors of p: 1,2,4,5,10,20

Factors of q: 1,2

Possible roots

1 ,2,4,5,10,20

± --------------------------------------------------------

1,2

So we get

±1, ±2, ±4 ,±5 , ±10 ± 20 ± 1/2± 2/2,±4/2,± 5/2,± 10/2,± 20/2

Simplifying

±1, ±2, ±4 ,±5 , ±10, ± 20, ± 1/2,± 1,±2,± 5/2,± 5,± 10

Eliminating repeats

±1, ±2, ±4 ,±5 , ±10, ± 20 ,± 1/2,± 5/2

Putting them in numerical order

± 1/2,±1, ±2,± 5/2, ±4 ,±5 , ±10, ± 20

Masteriza [31]3 years ago

3 0

Answer:

Its Option 1

Step-by-step explanation:

The possible rational roots will have a numerator that divides 20 (the last number) and a denominator that divides 2 (the coefficient of x^3).

For example 20/2, 10/2 and -1/2 = 10, 5 and -1/2.

The correct answer is the first option.

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Wewaii [24]

Answer:

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− 2 y ' ' − 4 y ' + 1 y = 0 : over-damped

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Step-by-step explanation:

Using the characteristic equation you can express a differential equation of order n as an algebraic equation of degree n:

$a_ny^n+a_n_-_1y^{n-1}+...+a_1y'+a_oy=0$

This differential equation will have a characteristic equation of the form:

$a_nr^n+a_n_-_1r^{n-1}+...+a_1r+a_o=0$

Now, you can classify the solution for a differential equation using a simple method. In order to do it, you just need to use the discriminant.

If the discriminant is greater than zero, the solution is over-damped

If the discriminant is less than zero, the solution is under-damped

If the discriminant is equal to zero, the solution is critically damped

So, given the differential equation:

$-3y''-3y+3y=0$

Which has characteristic equation of the form:

$-3r^2-3r+3=0$

The quadratic polynomial of the form:

$ar^2+br+c=0$

Has discriminant:

$Disc=b^2-4ac$

In this case:

$a=-3\\b=-3\\c=3$

So:

$Disc=(-3)^2-4(-3)(3)=9-(-36)=45$

In this case:

$Disc=45>0$

Therefore the solution is over-damped.

Now, given the differential equation:

$-2y''-4y'+1y=0$

Which has characteristic equation of the form:

$-2r^2-4r+1=0$

The quadratic polynomial of the form:

$ar^2+br+c=0$

Has discriminant:

$Disc=b^2-4ac$

In this case:

$a=-2\\b=-4\\c=1$

So:

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In this case:

$Disc=24>0$

Therefore the solution is over-damped.

Finally, given the differential equation:

$1y''+7y'+5y=0$

Which has characteristic equation of the form:

$1r^2+7r+5=0$

The quadratic polynomial of the form:

$ar^2+br+c=0$

Has discriminant:

$Disc=b^2-4ac$

In this case:

$a=1\\b=7\\c=5$

So:

$Disc=(7)^2-4(1)(5)=49-20=29$

In this case:

$Disc=29>0$

Therefore the solution is over-damped.

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Answer:

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Step-by-step explanation:

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