All categories

2 years ago

What is the quotient (6x4 − 15x3 + 10x2 − 10x + 4) ÷ (3x2 + 2)?

Mathematics

1 answer:

Alexandra [31]2 years ago

6 0

Answer:

The quotient of (6x4 + 15x3 + 10x2 + 10x + 4) ÷ (3x2 + 2)

is 2x2 + 5x + 2.

Step-by-step explanation:

You might be interested in

Use the Integral Test to determine whether the series is convergent or divergent

Inga [223]

Answer:

A. $\sum_{n=1}^{\infty}\frac{n}{e^{15n}}$ converges by integral test

Step-by-step explanation:

A. At first we need to verify that the function which the series is related ( $\frac{n}{e^{15n}}$ ) fills the necessary conditions to ensure that the test is effective.

*f(x) must be continuous or differentiable

*f(x) must be positive and decreasing

Let´s verify that $f(x)=\frac{n}{e^{15n}}$ fills these conditions:

*Considering that eˣ≠0 for all x, the function $f(x)=\frac{n}{e^{15n}}$ does not have any discontinuities, so it´s continuous

*Because eˣ is increasing:

if a<b ,then eᵃ<eᵇ

if 0<eᵃ<eᵇ ,then 1/eᵃ > 1/eᵇ

if 1/eᵃ > 1/eᵇ and a<b, then a/eᵃ<b/eᵇ

We conclude that $f(x)=\frac{n}{e^{15n}}$ is decreasing

*Because eˣ is always positive and the sum is going from 1 to ∞, this show that $f(x)=\frac{n}{e^{15n}}$ is positive in [1,∞).

Now we are able to use the integral test in $f(x)=\frac{n}{e^{15n}}$ as follows:

$\sum_{n=1}^{\infty}\frac{n}{e^{15n}}\ converges\ \leftrightarrow\ \int_{1}^{\infty}\frac{x}{e^{15x}}\ dx\ converges$

Let´s proceed to integrate f(x) using integration by parts

$\int_{1}^{\infty}\frac{x}{e^{15x}}\ dx=\int_{1}^{\infty}xe^{-15x}\ dx$

Choose your U and dV like this:

$U=x\ \rightarrow dU=1\\ dV=e^{-15x}\ \rightarrow V=\frac{-e^{-15x}}{15}$

And continue using the formula for integration by parts:

$\int_{1}^{\infty}Udv = UV|_{1}^{\infty} - \int_{1}^{\infty}Vdu$

$\int_{1}^{\infty}xe^{-15x}\ dx= \frac{-x}{15e^{15x}}|_{1}^{\infty} -\frac{-1}{15} \int_{1}^{\infty}e^{-15x}\ dx$

$\int_{1}^{\infty}xe^{-15x}\ dx= \frac{-x}{15e^{15x}}|_{1}^{\infty} -\frac{-1}{15}(\frac{-1}{15e^{15x}})|_{1}^{\infty}$

$\int_{1}^{\infty}xe^{-15x}\ dx= \frac{-x}{15e^{15x}}|_{1}^{\infty} -\frac{1}{225e^{15x}}|_{1}^{\infty}$

Because we are dealing with ∞, we´d rewrite it as a limit that will help us at the end of the integral:

$\int_{1}^{\infty}xe^{-15x}\ dx= \lim_{b \to{\infty}}(\frac{-x}{15e^{15x}}|_{1}^{b}-\frac{1}{225e^{15x}}|_{1}^{b})$

$\int_{1}^{\infty}xe^{-15x}\ dx= \lim_{b \to{\infty}} \frac{-b}{15e^{15b}}-\frac{1}{225e^{15b}}-(\frac{-1}{15e^{15}}-\frac{1}{225e^{15}})$

$\int_{1}^{\infty}xe^{-15x}\ dx= ( \lim_{b \to{\infty}} \frac{-b}{15e^{15b}}-\frac{1}{225e^{15b}})+\frac{1}{15e^{15}}(1-\frac{1}{15})$

We only have left to solve the limits, but because b goes to ∞ and it is in an exponential function on the denominator everything goes to 0

$\lim_{b \to{\infty}} \frac{-b}{15e^{15b}}-\frac{1}{225e^{15b}} = 0$

$\int_{1}^{\infty}xe^{-15x}\ dx= \frac{1}{15e^{15}}(1-\frac{1}{15})$

Showing that the integral converges, it´s the same as showing that the series converges.

By the integral test $\sum_{n=1}^{\infty}\frac{n}{e^{15n}}$ converges

7 0

3 years ago

Josie has an 18 square foot rug and a 24 square-foot rug her floor is covered in 1/3 square foot tiles how many tiles will the t

Hitman42 [59]

14 is the answer to this question

8 0

3 years ago

S use f(x)=2x-5 and g(x)=4x+3

jeyben [28]

Step-by-step explanation:

can you please add details to your question

7 0

3 years ago

Read 2 more answers

En un día de temporal de lluvia el meteorólogo dijo en las noticias que la probabilidad de que lloviera ese día era del 70%. ¿Qu

drek231 [11]

Answer:

Vamos a definir:

100% probabilidad de lluvia = Es seguro que ese día va a llover.

0% probabilidad de lluvia = Es seguro que ese día no va a llover.

Ahora, los valores intermedios admiten una dualidad, es decir, es posible que llueva o que no.

Particularmente en el caso del 70% significa que:

Si tenemos un dado de 10 caras, con los números {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Y lo tiráramos, los resultados {1, 2, 3, 4, 5, 6, 7} indicarían que llueve.

Los resultados {8, 9, 10} indicarían que no llueve.

Entonces es mas probable que llueva, pero la probabilidad de que no llueva no es despreciable.

5 0

3 years ago

The cross-sectional areas of a triangular prism and a right cylinder are congruent. The triangular prism has a height of 5 units

nydimaria [60]

Answer:

The volume of the prism is equal to the volume of the cylinder

Step-by-step explanation:

For each solid figure, the volume formula is ...

V = Bh

where B is the cross-sectional area and h is the height. The problem statement tells us B and h have the same values for both figures. Hence their volumes are the same.

5 0

3 years ago