The location of a dolphin in relation to the surface of the sea, h(x), over time, x, in seconds, for 5 seconds can be modeled by a cubic function. Each of the following functions is a different form of the cubic model for the situation given above. Which form would be the most helpful if attempting to determine the time it takes for the dolphin to re-enter the sea after leaping out of the water? h(x) = 2x2(x - 11) + 4(17x - 12) h(x) = 2x(x2 - 11x + 34) - 48 h(x) = 2(x - 1)(x - 4)(x - 6) h(x) = 2x3 - 22x2 + 68x - 48

The most helpful function in an attempt to determine the time it takes for the dolphin to re-enter is h(x) = 2·(x - 1)·(x - 4)·(x - 6)

Step-by-step explanation:

For 2·x²·(x - 11) + 4·(17·x - 12)

h(5) = 2×5^2×(5 - 11) + 4×(17×5 - 12) = -8

For the function h(x) = 2·x·(x² - 11·x + 34) -48 we have;

h(5) = 2×5×(5^2 - 11×5 + 34) -48 = -8

For the function h(x) = 2·(x - 1)·(x - 4)·(x - 6) we have;

h(5) = 2×(5 - 1)×(5 - 4)×(5 - 6) = -8

For the function h(x) = 2·x³ - 22·x² + 68·x -48 we have;

h(5) = 2×5^3 - 22×5^2 + 68× 5  - 48 = -8

Given that the values of the function are all equal at x = 5, the function that will be most helpful in determining the time it takes for the dolphin to re-enter the sea after leaping out of the water is the function that is already factorized

Thereby where the value of the function h(x) at which the dolphin re-enters the the sea is h(x) = 0, we have the function h(x) = 2·(x - 1)·(x - 4)·(x - 6), readily gives the time values, x, as x = 1 second or 4 second or 6 second, therefore, the most helpful function is h(x) = 2·(x - 1)·(x - 4)·(x - 6).

(2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

step-by-step explanation:

simplify the following:

x/x^2 + 3 x + 2 - (x (x + 1))/(x + 2)

combine powers. x/x^2 = x^(1 - 2):

x^(1 - 2) + 3 x + 2 - (x (x + 1))/(x + 2)

1 - 2 = -1:

x^(-1) + 3 x + 2 - (x (x + 1))/(x + 2)

put each term in 1/x + 3 x + 2 - (x (x + 1))/(x + 2) over the common denominator x (x + 2): 1/x + 3 x + 2 - (x (x + 1))/(x + 2) = (x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)):

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2))

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)) = ((x + 2) + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2)):

(2 + x + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

3 x^2 (x + 2) = 3 x^3 + 6 x^2:

(2 + x + 3 x^3 + 6 x^2 + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

2 x (x + 2) = 2 x^2 + 4 x:

(2 + x + 6 x^2 + 3 x^3 + 2 x^2 + 4 x - x^2 (x + 1))/(x (x + 2))

-x^2 (x + 1) = -x^2 - x^3:

(-x^2 - x^3 + 3 x^3 + 6 x^2 + 2 x^2 + 4 x + x + 2)/(x (x + 2))

grouping like terms, 3 x^3 - x^3 + 6 x^2 + 2 x^2 - x^2 + 4 x + x + 2 = (3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2:

((3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

3 x^3 - x^3 = 2 x^3:

(2 x^3 + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

6 x^2 + 2 x^2 - x^2 = 7 x^2:

(2 x^3 + 7 x^2 + (x + 4 x) + 2)/(x (x + 2))

x + 4 x = 5 x:

answer:   (2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

d, x=8

step-by-step explanation:

simplifying

4x + -5 = 2x + 11

reorder the terms:

-5 + 4x = 2x + 11

reorder the terms:

-5 + 4x = 11 + 2x

solving

-5 + 4x = 11 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-5 + 4x + -2x = 11 + 2x + -2x

combine like terms: 4x + -2x = 2x

-5 + 2x = 11 + 2x + -2x

combine like terms: 2x + -2x = 0

-5 + 2x = 11 + 0

-5 + 2x = 11

add '5' to each side of the equation.

-5 + 5 + 2x = 11 + 5

combine like terms: -5 + 5 = 0

0 + 2x = 11 + 5

2x = 11 + 5

combine like terms: 11 + 5 = 16

2x = 16

divide each side by '2'.