CPhill

First one

Avg rate of change on [ a, b ]  =  [ g(b)  - g(a)] / [ b - a]

So....on  [ -3, - 1]   we have  [ 9 - 5] / [ -1 - (-3) ] =   4/ 2  = 2

[-1, 1 ] is correct

[ 1, 3 ]  =   [ 13 - 17 ] / [ 3 - 1 ]  =   -4/ 2  =  -2

[ 3, 4 ] is correct

1.   If  x - a is a factor, then the remainder when P(x) is divided by x- a  will be 0

[ A "factor" of a polynomial means that it divides the polynomial evenly....there will be a remainder of 0 ]

2. What will be the remainder when the polynomial p (x ) = 3x 5 - 8x 3 + 4x 2 - 41 is  divided by x - 2?

If a polynomial  is divided by  x - a.....then the remainder   will =  P(a).....in other words....we put "a" into the polynomial  and evaluate.....the evaluation will be the remainder

So......x - a  = x -2      ....so....we put 2 into the polynomal  and we get

3(2)^5  - 8(2)^3  + 4(2)^2   - 41  =

3*32  - 8*8 + 4*4  - 41 =

96 - 64 + 16 - 41  =

7 =   the remainder

3. If the binomial (x-7) is a factor of the polynomial function f(x), which statement must be true?

If  x - a  is a factor....then "a"  is a zero....so  P(a) = 0

In this case.....x - 7 is a factor.....so......7 is a zero......so....P(7) = 0

Correct for the first one, guest....I missed the dot

No....Plane A travels farther in one hour than Plane B

First one.....if   a/b   is a root, then  (bx - a) is a factor

To see this   let  bx - a = 0

Then  bx = a

And  x  = a/b  is  the root

So....if 3/4 is a root, then,  (4x - 3)  is a factor

Second one

x^2 - 4 =  0     factor as

(x + 2) ( x - 2)  = 0

So....this will have two roots  (zeros)

Last one

If  x - a  divides P(x).....then  a is a  zero   (root)   ...so.....  P(a) = 0

So

If x - 2 divides P(x)......then 2 is a zero   ....so....P(2)  = 0

First one - last choice

Second one

Plane A travels 290 (1 hr)  =  290 miles  in one hour

Plane B  travels 540 / 2hrs  = 270 miles in one hour

Last one

Function A has  a vertical asymptote  where  x - 3 =  0  ⇒   at   x = 3

Function B has a vertical asymptote at x  = 4

It will be even

1*2*3*4*5  =  120  would be divisible by every integer from 1 - 10  inclusive except for 7 and 9

Multiplying this by 3  gives

1*2*3*3*4*5  =     1* 2*4*5*9  =  360

This will be divisible  by 1, 2, 3, 4, 5, 6, 8, 9, and 10......it will not be divisible by 7

Binomial Expansion  approach

(6701)^83  =  

(6700 + 1)^83  =  6700^83  + C(83,1)*6700^82 +  ...+ C(83,81)*6700^2 + C(83,82)*6700 + 1

Note that every term in red will have at least 4 trailing zeros, so  each of these terms is divisible by 1000

So....we only need consider the last two terms

C(83, 82) * 6700  +  1     =

83 * (6700)  + 1  =

(80 + 3) * (6700)  + 1 =

80*6700 + 3*6700  + 1 

The first term  will have 3 trailing zeros, so....it is divisible  by 1000....and

3 * 6700 + 1 = 20100  + 1 =  20(1000) +  100  +  1

So  we have  left

 [ 100 + 1 ] / 1000   ⇒    remainder 101

We can use a graph to help with this : https://www.desmos.com/calculator/4beobd4b3x

The vertex points of the quadrilateral  are  (0, 3)  (0,2)  (1,0)  and (3, 0)

By visual inspection , the longest side  is the distance from (0,3) to (3,0)

And this distance =  √ [ (3 - 0)^2 + (3 - 0)^2 ] = √  [9 + 9 ]  = √18  =  3√2 units