ChowMein

i posted as a guest, but thanks!

A' is the set of all elements that are in the universal set U, but not in set A.

So, the answer is c) {2, 4, 6}

You can use difference of squares: a² - b² = (a+b)(a-b)

[101² - 97²] + [93² - 89²]

(101-97)(101+97) + (93-89)(93+89)
(4)(198) + (4)(182)

(4)(198 + 182)    [using the distributive property: ab + ac = a(b+c)]
(4)(380) = 1520

The remainder theorem tells us that:

If a polynomial f(x) is divided by (x-a), then the remainder is f(a).

So for this problem, the polynomial is divided by (x-1), (x+1), and (x+2) and the remainders are given.

Therefore, we can conclude that

f(1) = 13

f(-1) = 3

f(-2) = -12

You can now plug in those x values and y values to solve for a, b, and c.

\(x^4+ax^3+bx+c\\ 1^4+a(1)^3+b(1)+c=13\implies a+b+c=12\\ (-1)^4+a(-1)^3+b(-1)+c=3\implies -a-b+c=2\\ (-2)^4+a(-2)^3+b(-2)+c=-12\implies -8a-2b+c=-28\)

We now have a system of three equations.

Now to find a, b, and c.

We can first add the first two equations together

\(a+b+c=12\\ -a-b+c=2\\ ==========\\ 2c=14\\ c=7 \)

We can then multiply the 2nd equation by -2 and add it with the 3rd equation

\(\underset{\downarrow}{-2(-a-b+c)=-2(2)}\\ 2a+2b-2c=-4\\ -8a-2b+c=-28\\ ============\\ -6a-c=-32\\ -6a-7=-32\\ -6a=-25\\ a=\frac{25}{6} \)

Now, we can plug in a and c into any other equation to find b.

\(a+b+c=12\\ \frac{25}{6}+b+7=12\\ \frac{25}{6}+b=5\\ b=\frac{5}{6}\)

TL;DR

a = 25/6, b = 5/6, c = 7

Each tick marks 60 units.

Samantha traveled two ticks to the right.

2 * 60 units = 120 units

Ryu traveled 8 ticks to the left.

(Negative values represent the distance traveled west, in this case to the left.)

-8 * 60 units = -480 units

Now there are two ways you can find the distance between them.

1) You can subtract their positions.

120 - (-480) = 120 + 480 = 600 units

2) Draw a picture.

As you can see from drawing the picture, the distance between them can be found by doing 480 + 120 = 600 units

that can be rewritten as this:
\(\sum_{n=1}^{\infty}\frac{2n+1}{n^2(n+1)^2}\)

so first we have to find the formula for the nth partial sum

\((1)\sum_{n=1}^{\infty}\frac{2n+1}{n^2(n+1)^2}\\ (2)\sum_{n=1}^{\infty}\frac{2n+1+n^2-n^2}{n^2(n+1)^2}\\ (3)\sum_{n=1}^{\infty}\frac{n^2+2n+1}{n^2(n+1)^2}-\frac{n^2}{n^2(n+1)^2}\\ (4)\sum_{n=1}^{\infty}\frac{1}{n^2}-\frac{1}{(n+1)^2}=\left(\frac{1}{1^2}-\frac{1}{2^2}\right)+\left(\frac{1}{2^2}-\frac{1}{3^2}\right)+\left(\frac{1}{3^2}-\frac{1}{4^2}\right)+...+\left(\frac{1}{n^2}-\frac{1}{(n+1)^2} \right)=1-\frac{1}{(n+1)^2}\\\)

step 2: try to turn the series into a more recognizable telescoping series 

step 3: split the fraction

step 4: the terms cancel out leaving 1-1/(n+1)^2

now we take the limit to infinity of the nth partial sum formula to find the value

\(\lim_{n\rightarrow \infty}1-\frac{1}{(n+1)^2} = 1\)

since you're in algebra another way to look at the thing above is as n gets bigger and bigger to infinity, 1/(n+1)^2 will become 0, leaving 1-0 = 1

the problem is your 3rd step
 

(8-i√11) (8+i√11) = 
64 - i²(11)

that should get you 75

 

a)
Jesse could raise the temperature by 10 degrees Celsius.

b)
-6.2°C + 10°C = 3.8°C

\(\begin{bmatrix} -4 & 1\\ -1 & 2 \end{bmatrix} \begin{bmatrix} -9\\ 2 \end{bmatrix}= -9 \begin{bmatrix} -4\\ -1 \end{bmatrix}+ 2 \begin{bmatrix} 1\\ 2 \end{bmatrix}= \begin{bmatrix} 36\\ 9 \end{bmatrix}+ \begin{bmatrix} 2\\ 4 \end{bmatrix}= \begin{bmatrix} 38\\ 13 \end{bmatrix}\)

the ship is less dense than the water, hence why it floats

density is the amount of mass that's packed into a given volume or space.

the ship has less density than the water because it takes up a large amount of space
therefore it floats

x = -0.2t³ + 0.06t² + 10 [m] 
v_x = x' = -0.6t² + 0.12t [m/s]
a_x = v_x' = -1.2t + 0.12 [m/s²]
a_x(1) = -1.08 m/s²

ΣF_x = ma_x = (180kg)(-1.08m/s²) = -194.4N

y = 0.1t² + 0.4t + 6 [m]
v_y = y' = 0.2t + 0.4 [m/s]
a_y = v_y' = 0.2 [m/s²]

ΣF_y = ma_y = (180kg)(0.2m/s² + 9.8m/s²) = 1800N

y=3-x is a line, so the slope is going to be the same everywhere on that line

we can pick two points on the line
x₁ = 0, y₁ = 3
x₂ = 1, y₂ = 2
(you could pick any two points)

slope = (y₂ - y₁) / (x₂ - x₁) = (2 - 3) / (1 - 0) = -1
slope = -1