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x^3 - x^2  -4x + 4

f(-1)  =   (-1)^3  - (-1)^2 - 4(-1) +  4  =  6  ....so...first one is not true

If the second is true then

f(1)  = 0  ...  (1)^3 - (1)^2  -4(1) + 4 =  0  .....so....the second is true

Since x = 1  is a root....then the 3rd one is true 

The 4th  means the same thing as the first......

So...the 2nd and 3rd statements   are true

We can do this with something known as "Pick's Theorem"

Here's a pic  :

Area  =

Number of Interior Lattice Points  + Number of Border Lattice Points / 2   - 1

Where a "lattice point"  is a point with integer coordinates  [ in our case....the intersection of two gridlines]

Number of interior Lattice Points  = 24

Number of Border Lattice Points  =  6

So....the area is

24  +  6/2  + 1   =

24 + 3  - 1  =

26 units^2

Huh I guess this was a harder problem than I thought!  Sorry for the impatience, it was just due yesterday so I got really antsy.  Thanks for the help everyone!

Here's the parent and the transformed function :

https://www.desmos.com/calculator/yml1zmv3vi

We can do this with synthetic division

4   [  1     - 5     7     -12   ]

                 4    -4      12

        _________________

        1     -1      3       0

The remaining polynomial  is   x^2 - x  + 3

a)  Call the consecutive free throws needed, x

So we have

[ 12 + x ]  / [ 15 + x ]  =  .85

b)   Multiply both sides by  15 + x

12 +  x   =  .85 ( 15 + x)     simplify

12 + x  =  12.75  + .85x      rearrange

1x - .85x   =  12.75 - 12

.15x  =  .75      divide both sides by  .15

x  = 5   [ consecutive free throws ]

[x - 4 ]| = x^2 - x +  3 [Quotient]

x^3 | - | 5 x^2 | + | 7 x | - | 12

x^3 | - | 4 x^2 | | | |

               -x^2 | + | 7 x | |

               -x^2 | + | 4 x | |

                             3 x | - | 12              

                             3 x | - | 12

                                     0

If I've interpreted this correctly.. {it's a little hard to see ]

Here you go :

https://www.desmos.com/calculator/ykscvtbsbm

There is only one x /y intercept  at   (0,0)

1st  way....note that a 15% discount  is the same thing as multiplying  by .85

Take 5 dollars off the original price = p -5

Take a 15% discount on this  = (p - 5) .85

(p - 5).85   =  .85p - 5(.85)  =   .85p - 4.25  = final price  (1)

Second way

Apply the discount   .....then the coupon

.85p  - 5  = final price  (2)

If you can see this .....subtract (2) from (1)....so we have

.85p  - 4.25  -  ( .85p - 5)

-4.25  + 5  = 

$ [ 5 - 4.25]  = 

$.75  more

2.345 x 10^6 = 2,345,000

Simplify the following:

(x^2 + 3 x - 28)/(x^2 - 7 x + 12)

Factor x^2 - 7 x + 12 by finding two numbers whose product is 12 and whose sum is -7.

The factors of 12 that sum to -7 are -3 and -4. So, x^2 - 7 x + 12 = (x - 3) (x - 4):

(x^2 + 3 x - 28)/((x - 3) (x - 4))

Factor x^2 + 3 x - 28 by finding two numbers whose product is -28 and whose sum is 3.

The factors of -28 that sum to 3 are 7 and -4. So, x^2 + 3 x - 28 = (x + 7) (x - 4):

((x + 7) (x - 4))/((x - 3) (x - 4))

Cancel common terms in the numerator and denominator of ((x + 7) (x - 4))/((x - 3) (x - 4)).

((x + 7) (x - 4))/((x - 3) (x - 4)) = (x - 4)/(x - 4)×(x + 7)/(x - 3) = (x + 7)/(x - 3):

=(x + 7) / (x - 3)

Simplify the following:

5/(1/6 + 1/(x + 1))

Put the fractions in 1/(x + 1) + 1/6 over a common denominator.

Put each term in 1/(x + 1) + 1/6 over the common denominator 6 (x + 1): 1/(x + 1) + 1/6 = (x + 1)/(6 (x + 1)) + 6/(6 (x + 1)):

5/((x + 1)/(6 (x + 1)) + 6/(6 (x + 1)))

Combine (x + 1)/(6 (x + 1)) + 6/(6 (x + 1)) into a single fraction.

(x + 1)/(6 (x + 1)) + 6/(6 (x + 1)) = ((x + 1) + 6)/(6 (x + 1)):

5/((x + 1 + 6)/(6 (x + 1)))

Write 5/((x + 1 + 6)/(6 (x + 1))) as a single fraction.

Multiply the numerator of 5/((x + 1 + 6)/(6 (x + 1))) by the reciprocal of the denominator. 5/((x + 1 + 6)/(6 (x + 1))) = (5×6 (x + 1))/(x + 1 + 6):

(5×6 (x + 1))/(x + 1 + 6)

Group like terms in x + 1 + 6.

Grouping like terms, x + 1 + 6 = x + (1 + 6):

(5×6 (x + 1))/(x + (1 + 6))

Evaluate 1 + 6.

1 + 6 = 7:

(5×6 (x + 1))/(x + 7)

Multiply 5 and 6 together.

5×6 = 30:

(30 (x + 1))/(x + 7) =(30x + 30) / (x + 7)

a. The fundamental theorem says that a polynomal of degree "n"   will have "n" roots

Note that some may be repeated  and/or some may be complex

So.....this a polynomial of degree 2....so...it will have two roots

b.

2x^2 + 4x + 7  =  0      

Use the quadratic formula   a = 2, b = 4, c = 7

So we have

x_1,2  =  [  - 4 ± √ [4^2 - 4(2)(7) ] ]  /  [ 2 * 2 ]  =

[  -4 ± √ [ 16 - 56]  ]  / 4  =

[ -4 ± √ -40 ] / 4  =

[ - 4 ± 2i√ 10 ] / 4  =

[ -2 ± i√ 10 ] / 2

This is a trick question ,NSS

Note what this says

(  g^-1  °  g ) (5)    means that we first put "5"  into "g"

We will get out  some value  "k"

So......we have the point 

(5, k)

Lastly....we put this result into the inverse....

But  the inverse just " reverses"  the original coordinates....so... putting "k" into the inverse produces

(k, 5)

So....the final answer is  just what we started with ⇒  "5"

Do you see this???....

First...let's get the transformations for this....this is always helpful...

The 'parent function " is the square root function

The negative out front reflects it across the x axis

The  "0.5 "  vertically compresses this by a factor of 1/2

The  " (x - 3) "   shifts this 3 units to the right of the parent function

Finally....the    "+2"  shifts this up 2 units

Here's the parent  and the transformed functions :

https://www.desmos.com/calculator/srnse7dgs4

7)

Solve for x:

75000 (x - 1940)^(1/3) = 245000

Divide both sides by a constant to simplify the equation.

Divide both sides by 75000:

(x - 1940)^(1/3) = 49/15

Eliminate the cube root on the left hand side.

Raise both sides to the power of three:

x - 1940 = 117649/3375

Solve for x.

Add 1940 to both sides:

x = 6665149/3375 = ~1975

We can just solve the equations inside the parentheses

So we have

-9x + 5  =  3 + 4x    add 9x to both sides, subtract 3 from both sides

2 =  13x       divide both sides by  13

2/13  = x

f / g  =   [ 3x - 6 ] / [ x - 2]   =   3[ x - 2 ] / [ x - 2 ]  =  3

Note....that we can't put 2 into the original function because it makes the original denominator = 0

So.....second answer

We  can fing this using some trig

We have a right triangle  with one leg  = y/2

And the angle   opposite this side = 22.5°

And   1/2 of the distance between the parallel sides is the other leg....and the angle opposite this side   is  67.5°

So....using the Law of Sines....we can find half of the distance as

(y/2)sin (67.5)  / sin (22.5)  =  (y/2) ( 1 + √2)  =  [ ( 1 + √2) / 2 ]   y

So...twice the distance is   (1 + √2) y  = distance between the parallel sides

I got the following numbers:

Base of the small triangle with 36-degree angle = 8.7831 cm, Height =6.38129 cm

Base of the large triangle with 20-degree angle =28.7831 cm, Height= 10.47619 cm

Volume of a cone  =

(1/3) (3.14) (diameter /2)^2  (height)

(1/3) (3.14) (18/2)^2 (18)  =

(3.14/3) * (9)^2 (18)  =

3.14 *  81  * 18/3   =

3.14 * 81 * 6  ≈    1526.04 ft^3

Since zero is not allowed at the beginning of each set of 4 ascending numbers, then the total number of 4-digit sets would simply be: 9C4 =126.

Using your diagram, I can prove that  AX  = BY

Note that we have two triangles, ACX  and BCY

AC    =  CY   =  "b"

And

XC  = BC  =  "a"

And  angle ACX  = angle YCB

So.... by SAS   .... triangle  ACX  is congruent to triangle YCB

So... AX  =  BY

But...I have not figured out how to prove that CZ  is also equal to these

Maybe someone can provide the final step  ???

36a^4b^10 - 81a^16b^20   =

(6a^2b^5  + 9a^8b^10) ( 6a^2b^5 - 9a^8b^10)